“You know,” says Mr Germanotta of his elder daughter, “she won a jazz competition when she was about 14 or 15. That was her roots: she was trained in jazz. It’s a great foundation for singing.”

Another foundation was supplied in the family home. “Oh sure I played Tony Bennett at home,” adds the hearty proprietor, as proud an Italian New Yorker as they come. “Especially when you’re in the restaurant business, you put him on all the time. Tony Bennett is an old American staple.”

That he is. A star of a bygone age and the last of his era, 88-year-old Anthony Dominick Benedetto recorded his first single, Fascinating Rhythm/ Vieni Qui, in April 1949, in New York’s Decca studios. His 14th single, 1951’ s Because of You, gave him his first hit, selling a million copies. The same year he was headlining the city’s Paramount Theatre, playing seven shows a day. He hasn’t stopped since, releasing – in Joe’s estimation – “300 albums or something like that”. Friend, peer and rival Frank Sinatra paid him the ultimate compliment: “Tony Bennett has four sets of balls.”

And then there’s Lady Gaga. The quintessential modern pop star, with her 42 million Twitter followers – or, Little Monsters, as they proudly and devotionally style themselves – myriad costume changes ( even offstage, and often in the same day), outrageous wigs, over-the-top concert spectaculars, admissions of emotional and narcotic weakness, and dresses made of meat.

When Bennett first heard her, he confidently predicted she’d be “bigger than Elvis”. He also called her “the Picasso of pop”, a claim that holds slightly more water than it might otherwise given Bennett’s status as a keen painter.

“Tony heard her, they did something together, and then he said, ‘ Let’s do a whole collaboration,’ ’’ recounts a pleased-as-punch Mr Germanotta. “And I’ve been sitting in the studio and it’s just been incredible. They have chemistry.”

As much is evident 20 blocks and three hours later. In an uptown recording studio Lady Gaga and Tony Bennett are sitting down together. Five songs from that “whole collaboration”, a covers album called Cheek to Cheek, is today being unveiled through giant speakers. Well, they’re unveiling five of their faithful covers of standards culled from the Great American Songbook.

That simpatico relationship is also evident on Gaga’s flesh. True to form, she’s put her art on her sleeve – a Bennett sketch of Miles Davis’s trumpet, including the artist’s signature (“Benedetto”), is tattooed on her arm. The skin is still red, raw and angry-looking. Presented close-up with this evidence of her devotion, I swear the old-school singer blanches a little.

The jumping-off point for the project was “something they did together”: a recording of The Lady is a Tramp, which opened Bennett’s 2011 album Duets II. It was a crowded compilation that had the elder statesman partnering with 16 other singers, including Amy Winehouse, Michael Bublé, Aretha Franklin and k. d. lang – and was itself the sequel to 2006’ s Duets: An American Classic, which featured 19 singers (Bono, Sting, Juanes, some people with two names). But on Cheek to Cheek, both the song and the album, it only takes two to tango.

“Well, that was the first record we talked about actually,” begins Gaga of a song written for the 1935 Fred Astaire/ Ginger Rogers movie Top Hat. Those are big shoes to fill, and to her credit – on one listen anyway – Gaga does a genre-appropriate and respectful job, sounding a little like Dinah Washington. “This is the Great American Songbook,” she continues. Her New York accent is strong, but she seems to be dialling down, or holding back, her normal impassioned speaking style in deference to a man of whom she’s clearly in awe. Gaga is dressed demurely, for her: today’s wig is one colour (black) and one length (long).

Bennett, sitting within hand-holding distance in an adjacent armchair, is smartly suited and coiffed. She smiles reverently at him before carrying on. “It’s meant to be sung till the end of time. And Cheek to Cheek is the most simple and most beautiful way you can describe this relationship. Nothing matters except that we are together and we are having a conversation with jazz.” This pair of New York-Italians first met at a 2011 hometown gala benefit hosted by the Robin Hood Foundation. What did Bennett know or think of Lady Gaga before that encounter? He seems not to hear or understand my question, which may be as much to do with my accent as anything. He replies in his low, gravelly, methodical speaking voice that, “it was the first time I ever saw her perform. I could not believe the audience reaction to her.

“I’ve been around long enough to know what every artist does. But this audience – it wasn’t just that one night, but every time I see her… and it’s not just little tiny children. She has an audience of all ages that are singing the songs that she performs. And they’re all singing in unison with her. So the audience just adores her.”

Since her emergence five years ago, Lady Gaga has had titanic success. Her third studio album, last year’s Artpop, may not have met with the same critical and commercial acclaim as its predecessors The Fame and (to a lesser extent) Born This Way. But 200 million record sales don’t go away overnight, and she’s still a huge global force; next month her global Artrave tour comes to the UK’s arenas.

Still, for all her seemingly Teflon-coated confidence, she’s no pop autobot. She admits that when she first went into the studio to record The Lady is a Tramp, her nerves were such that she resorted to a shot or two of whisky.

“Well of course I was so nervous!” Gaga retorts when I mention this. “Yeah, but you showed up so prepared,” soothes Bennett. “Well that’s ’ cause I was nervous!” she repeats, laughing. But she wasn’t drunk? She shoots me an as-if look. “No, it takes more than a few whiskys to take me down, I’ll tell you,” snorts this woman with a predilection for partying. “But of course I was nervous, he’s so…”

“You know what?” interjects Bennett. “Every great performer I’ve ever met, they’re extremely nervous because they care so much. They want to make sure the audience gets the whole thing. Am I gonna remember all the words? Are the musicians gonna play with me so that it’s natural? Are the lights right? Everything. You just wonder what’s gonna happen before you hit that stage.”

He knows of what he speaks: despite his multiple cojones, Bennett used to vomit with fear before performances or recordings.

“Now the artists don’t feel that nervousness,” he offers sagely, “there’s something flat that happens when they walk out. There’s a lack of energy. The audience sees that so they just…” Bennett mimics a desultory, bored clap.

“But another performer with that live energy, right away the audience responds to it… And the best artists – Sinatra, Lena Horne, the greatest artists that I’ve ever met and watched and performed [ with], were the ones – Ella Fitzgerald!” he exclaims, “before she hits the stage she almost forgets what her name is. That’s how she works.”

I ask Gaga about the jazz contest that her father had mentioned. “The all-state jazz competition,” she nods. “It was students all over New York state. You win within your sector, then you got to go to this convention upstate in New York where everybody would compete for all sorts of medals and beautiful plaques. And I was the only person chosen from my school and I was so proud and happy to go.”

“And it was always a great experience for me to be around people that loved music, ’ cause I didn’t have a great performing arts programme at my school. That’s why I love Tony so much ’ cause he has all those performing arts programmes and he cares so much about education and art.”

In 2001 Bennett opened the Frank Sinatra School of the Arts (FSSA) in his home neighbourhood of Astoria in Queens. Four years ago the programme was expanded to seven public schools in New York. Earlier in the week of our meeting, the pair had surprised 700 students at the FSSA with an appearance in the Tony Bennett Concert Hall, fielding questions and singing The Very Thought of You (him) and Every Time we Say Goodbye (her). Having heard (once) her versions of Anything Goes, Mad About The Boy and It Don’t Mean a Thing (if it Ain’t Got That Swing), I can attest that Gaga does seem to have been having a whale of a time singing these standards. And as Bennett affirms, “boy can she belt ’ em out”.

But whence that passion? In a press statement released when the album was announced, Gaga was quoted as saying: “It’s liberating to be singing jazz, and especially at this point in my career.” How so and why “especially” now?

“Well, it’s been extremely liberating for me ’ cause jazz is rebellious. And I’m a rebel at heart. So in that way it’s also me rebelling against my own pop music – that’s really exciting for me. But I think the thing that has been the most powerful has been, there’s a part of me that has been quiet for a long time that is now being reawakened, after years of producers and record label people telling me to make my voice sound more radio-friendly.

“Since The Fame, The Fame Monster and Born This Way, they’ve been auto-tuning it more, or changing the timbre. They take the vibrato out so you sound like a robot.”

Before I can step in with a question – viz, “is that the truth? A major artist of your widely reported control?” – Tony’s beat me to it. “Jeez!” he says. “… they really control you,” she nods, “especially in the beginning. Although it was still my songs, and I still had a lot to say about the production, the vocal was something that they really, really wanted to control. So my vocal presence has been kind of the smallest presence about me for a long time. So everything else becomes the focal point.” Enter the jazz cavalry. Eyes blazing, she affirms that the Cheek to Cheek project, and specifically Bennett, have “let a wild animal out of a cage…”

“So when I say ‘ at this point in my career’, he’s changed my life for me because he’s gonna make my art better from this moment on. Because he said: ‘ Hey, you’re so much better than you’re even letting yourself be.’ ”

Does Bennett agree, that she wasn’t reaching her full vocal potential on her pop records? He replies that she “needs to be left alone and she should sing how she feels”. Sinatra, Ella, Nat King Cole – they all did that. He, then, is aligning her with those 20th-century vocal greats. Gentlemanly unimpressed as he is with social media stats and tabloid kerfuffles, he seems able to see past all the bells and whistles and wigs and steak frocks and focus on the singer within.

“See, it’s so funny,” chips in Gaga. “I think you’re right – a lot of people don’t know where I am because it’s so, um, eccentric to them, the way that I am. But he just sees an artist. And he hears the voice. I think the way I am is actually more old school than people realise.”

Our carefully calibrated time is nearly up. The duo are doing only a handful of interviews worldwide. But two days ago they undertook a nine-hour shoot with fashion photographer Steven Klein and, well, Bennett is both legend and old man, one with his own painting and concert schedule. Plus, Gaga’s world tour extravaganza restarts soon. Time with the pair of them, then, is at a premium. Before they go, I ask about Lush Life. There’s a weariness in Gaga’s voice as she sings the Thirties standard: “I used to visit all the very gay places, those come-what-may places, where one relaxes on the axis of the wheel of life, to get the feel of life from jazz and cocktails…” And: “life is lonely again, and only last year, everything seemed so assured, now life is awful again…”

Might that song be a depiction of the last couple of years of Gaga’s life? Among other setbacks, a hip injury forced her to cancel the end of her Born This Way tour, she split with her long-standing manager, and she admitted to heavy use of marijuana – seemingly it took 15 to 20 joints a day to cope with physical and mental distress.

“It is an autobiographical song,” she affirms. “And I said that to Tony the second I sang it to him in the studio. And I was crying and he put his arms around me and I said, ‘ Tony, I just feel like I’m such a mess, and I don’t want to be a mess for you.’ And he said, ‘ You’re not, you’re a sophisticated lady!’ Then he sang that song [Sophisticated Lady] back to me.

“That song,” she goes on, referring again to Lush Life, “I sang it when I was 13 years old. Mr Phillips, the jazz director at the Regis High School, gave that song to me and said: ‘ You could sing the hell out of this song’. And I didn’t know what it was about, but it’s like I sang it into reality, into existence. Singing that song with Tony, putting that down – gosh it was like therapy, you know?

“And,” she adds, “I don’t like to say that about music ’ cause it’s not really therapy – I really love to do it. But that was really a therapeutic moment for me: to revisit singing jazz and then letting out all of that… the pain of fame.”

“The pain of fame,” murmurs Bennett, roused by the default talkative passion that Gaga has clearly been stoppering thus far through deference to her partner. “That’s good,” he chuckles.

Has making Cheek to Cheek changed her as an artist and a person? “Yes. Absolutely. He has made me so happy in the way that I really needed to feel. He’s really, like, saved my life.”

Really? “Yeah, he really has. ’ Cause I really love music, and I really love being a singer, but I really hate being put in a box. And when I get put in a box I get very like a wild animal. And he let me be free. And I get to be with him while I’m doing it, and he’s teaching me all these life lessons, and I’m singing music I’ve loved my whole life. There’s no better music than the Great American Songbook. There just isn’t.

“And I feel a sense of confidence about myself as a musician that I always felt in the beginning. And then now I’m like: that’s right, you’re a cat. You got this. This is who you are. And,” she beams, looking headlong at Uncle Tony and clutching his hand, “it’s ’ cause I got him by my side.”

To return to Sinatra’s famous description of Bennett: Tony, how many sets of balls does Lady Gaga have? He gives a raspy laugh and outs her: “About 10!” Gaga laughs, delighted. “Don’t tell them all my secrets, Tony!”