The singer approached GaGa backstage at a New York charity concert where he asked her about appearing on the record.
"She was so warm and friendly and sweet. I said, 'I'd love to make a recording with you'. And she said, 'Anything you want, I'll do it," Bennett told The Guardian.
"I was thrilled by her reaction, because she's in such great demand, so I didn't know what her response would be... Some people are on tour or can't make the date, so you never know."
Their recorded version of 'The Lady Is a Tramp' features on the 18-track Duets II, which also includes songs with Amy Winehouse, Mariah Carey and kd lang.
Bennett added: "Well, each artist was a knockout. Each one shows they're going to be permanent parts of our lives."
He has previously suggested that GaGa will one day become as famous as Elvis Presley.
> Lady GaGa is most talented artist I've ever met, says Tony Bennett
Watch Lady GaGa sing 'Happy Birthday' for Tony Bennett below: