The New Yorker Bennett, 70, who came to fame in the 1950s with hits such as "Because of You", and "Cold, Cold Heart", saw his career decline in the early 1970s but come back to life again in the past five years.
Bennett won four Grammys for his acclaimed MTV Unplugged appearance and has become a cult hit with a new generation of fans.
Bennett even co-presented the MTV music awards with the Red Hot Chilli Peppers.
Speaking in London, Bennett said: "I met Sting and Paul McCartney and it was so exciting because they wanted to know how I'd survived all these years. They've had great success and want it to last. They asked me 'How do you do it? How do you stay popular all these years?' "
His advice? "I just told them to have fun and enjoy themselves. It was a great treat and a great surprise to find out they respected me."
Now he is hoping to record with them.
He said: "I'd like to do something with Sting and Paul one day. They're so good and they're very intelligent about how they go about things."
Bennett's new album is called Tony Bennett on Holiday, a tribute to Billie Holiday.