The singer had been reluctant to make experimental track "Carnival of Light" public after fellow band members - John Lennon, George Harrison and Ringo Starr - considered it too adventurous.
But now the star is planning to get permission from Ringo, who is the only other surviving Beatle, John's widow Yoko Ono and George's widow Olivia to release the 1967 record.
Paul told BBC Radio 4 arts program "Front Row": "It does exist. We were set up in the studio and would just go in every day and record."
"I said to the guys, this is a bit indulgent, but would you mind giving me 10 minutes?"
"All I want you to do is just wander round all the stuff and bang it, shout, play it. Then we put a bit of echo on it. It's very free."
The show's presenter John Wilson later revealed how Paul had told him privately how he would "love" to release the 14-minute number.
The track features lots of psychedelic sounds while John and Paul scream random phrases including "Are you all right?" and "Barcelona!."
