At a hotel in London, one half of the most popular duo in rock history is apologising for keeping me waiting. No worries, I tell him. I don't have anything else on, apart from an interview for the JC later in the day with comedian Jackie Mason. "Isn't he fabulous?" Art Garfunkel says with a smile, taking a seat in the bar. "I'm wild about that guy. Please tell him that I bow to his extraordinary talent. He's the real thing."

I venture that Mason defines Jewishness for a lot of people. Does the same go for the singer born Arthur Ira Garfunkel, owner of the world's most recognisable Jewfro (even if, aged 73, it is fast receding now)?

"I hope not," he says. "I like to think I sing for the universal spirit. I'm not in favour of identifying with religious differences. I love the Jews. They're bright, they're motivated, they're pushy. But I sing for anybody with a heart and sense of beauty."

His all-embracing attitude has served him well. Like those other Jewish musicians who became household names in the '60s and '70s - including Bob Dylan, Leonard Cohen, Lou Reed and, of course, Paul Simon - Garfunkel spoke to a generation, irrespective of their creed. He's a singer first, and a Jew second, which is why he is a little uncomfortable with my opening gambit: to enquire whether his barmitzvah in 1954, during which he sang as a cantor, was effectively his live debut?

"What a first question!" he says, laughing. "So I'm a Jewish singer? I think of myself as a singer."

He has been a rather successful one. Simon and Garfunkel sold tens of millions of records, including 25 million for 1970's Bridge Over Troubled Water, which, until Michael Jackson's Thriller, was the biggest-selling album of all time.

And it all began, not when he was 13 years old, but when he was nine, at a performance of Alice In Wonderland at his school in Queens, New York, in which Garfunkel, playing the Cheshire Cat, met Simon, the White Rabbit. The latter was impressed by the former's voice, and the effect it had on girls. As for Garfunkel, he fell for Simon's sense of humour.

"He was really a great joker," he recalls as we move from the somewhat noisy bar into the sunny courtyard outside. "He was kind of like Lenny Bruce - tearing hypocrisy off of the American cliché. And it brought out my funny side. I'm not bad myself, I have a good ear for the absurd. So we kept each other going. We became each others' pals. And soon we started being musical together. There was nobody round us on our level. We were the turned-on kids in the neighbourhood."

By 1957, influenced by the Everly Brothers, the pair were harmonising teens operating as Tom & Jerry, enjoying a minor hit with Hey, Schoolgirl. Then, suddenly, they fell out, and they didn't speak for five years.

What happened? Was it over a woman? "No," Garfunkel says. "He made a record without telling me and I was shocked and surprised. He was my best friend and I thought friendship involves candour and telling a friend what's up, so the fact that he had a hidden surprise hurt me."

I wonder whether the relationship recovered, and he replies: "Not really." It is a strange thing to say about a partnership that went on to bequeath some of the monumental releases of the counter-cultural era, particularly The Sound Of Silence, Homeward Bound, I Am A Rock and Bridge Over Troubled Water. Then again, they were feuding long before Liam and Noel Gallagher were born (the disputes were over musical control - Simon was the songwriter, Garfunkel the singer, with some say over production and arrangement). Was this a sign of fissures to come?

A weary smile: "Yes." It is interesting, I note, that one of music's iconic unions was essentially flawed, and doomed to fail. "Oh, it's fascinating - worth a whole book," he says, and it's hard to tell as he squints in the sun whether he's being facetious.

He marvels at the serendipitous nature of their meeting: "How compatible our different talents are, the coincidence of living three blocks from each other, the fact that Paul was so great on guitar and the popularity of my singing, this wrapping around each others' gifts… At such a young age, it stamped us in a deep way as compatible talents."

It is a wistful reminiscence, which figures: at the height of the '60s party, Simon and Garfunkel's songs provided a melancholy contrast to all the upheaval and tumult. "Good point," he says. "Paul's poetic style was so advanced, with its shades of dark grey. He had an unusually sophisticated poetic gift."

Simon and Garfunkel went their separate ways in 1970. He starred in two Mike Nichols films, Catch-22 and Carnal Knowledge. Then, in a surreally unexpected move, the educated Garfunkel (he had a BA in art history and MA in mathematics) took a job as a maths teacher at a private school in Connecticut, encouraged by his fiancée Linda Marie Grossman. There he was, one of the biggest stars on the planet, attempting to teach calculus to unmotivated teenagers.

"You do this kind of walking-away-from-it-all gesture sometimes," he suggests. "When you're sitting on top of the world and you've had enough of the noise and the glory and you want a rest from Paul Simon - enough, already - you imagine that the country and not the city is where you want to be, that this woman you met, Linda, is somebody to marry [they divorced in 1975], that a cottage in the country might be good…

"I'd say to the class, 'We're not going to talk about Bridge Over Troubled Water, we're going to talk about geometry and at the end of the year I'll deal with the fame trip,'" he explains. "And they continued for the rest of the term with, 'He's really talking geometry!'"

After nine months, he realised this was "a forced concept" and he returned to showbusiness, which he found "much more dynamic after the pettiness of the school teacher's life".

He recorded several solo albums, notably Angel Clare (1973) and Breakaway (1975), and had a UK No 1 in 1979 with Bright Eyes, as featured in the film version of the novel Watership Down. Tragically, that same year, his longtime girlfriend, actress Laurie Bird, committed suicide at their Manhattan apartment while he was filming his role in Nicolas Roeg's Bad Timing. It took him a long time to recover.

"It took half of the 1980s," he admits. "I was suitably shocked. I became very reclusive and introverted while I dwelt upon how great Laurie Bird was. People would say, 'Yes, but she's gone.' And I would say, 'I also think about how great J S Bach was, and he's gone, too.'"

Carnal Knowledge, Bad Timing and Boxing Helena in 1993 - many of Garfunkel's starring roles were in movies about sexual obsession.

"Oh yeah, interesting point," he says, grinning. Did they pick him, or did he pick them? "I am a Scorpio, Paul," comes his oblique response. "Maybe they're on to me."

Then he furthers, more revealingly: "I'm a bit of a momma's boy - I had the sweetest mom, and I was raised middle-class. I might be working against such a sweet suburban background when I look to play the disturbed killer. It's counterpoint. I do this time and again."

Another celebrity with distinct on- and offscreen personas is Jack Nicholson, with whom he acted in Carnal Knowledge. "He's not a wild man at all, he's a funny man, a devoted artist, a real Jersey guy, who happens to love women," he says. "He's terrific, with a medieval code of honour - he's honourable and loyal."

Paul Simon, on the other hand, Garfunkel said recently that his relationship with him "caused me more pain that my relationships with anybody else I could name".

"I said that? That seems so flamboyant. Some journalist pulled that nonsense out of me," he retorts, feathers ruffled. "That's not me. Paul Simon is a man who has enormously enriched my life, period."

And vice versa?

"We've both had a great effect on each others' lives," he says. "What would have been his life if his friend Artie didn't sing so good and help produce those records so good? What would it have been? Something smaller."

He does, however, accept that it is a relationship notorious for conflict.

"It's quite imperfect, and the imperfection makes you miss that they couldn't make more albums," he decides, repeating the sort of appeals he has heard for decades, ever since 1970. "'Can't you guys get it together? We love those albums!' I see it, I get it, I share it. I'm a huge Simon and Garfunkel fan. Tell that to Paul! I want to see them get it on. There's a sixth album in them."

Does he agree that the pressure is almost too great to match the success of the previous five?

"That's what Paul says. But I don't see it that way. My respect for Paul Simon is undimmed here in 2015," he says, imagining more of those appeals from the public. "'Get it together, boys! What's your problem?' I'll tell you the problem: Paul Simon is shadow-boxing against Simon and Garfunkel."

Garfunkel's advice to Simon? Stop kvetching about past laurels and get in the studio.

"Never mind the pressure of what you're supposed to do," he counsels. "You just have to take it step by step."

Art Garfunkel’s UK tour starts in September. For more information www.livenation.co.uk/artist/art-garfunkel-tickets.