The singer explained that Richards was envious that the Beatles had four good vocalists compared to the Stones' one.
"I talked to Keith Richards a couple of years ago, and his take on it was: 'Man, you were lucky, you guys, you had four lead singers,' whereas The Rolling Stones only had one," McCartney told the Radio Times.
"We were an entity. Mick [Jagger] used to call us the four-headed monster. We would show up at places all dressed the same."
McCartney also suggested that The Beatles were lucky to have got together as they missed compulsory national service duty by a few years.
"One of the most amazing things for The Beatles is that we just missed it," he added.
"A couple of years earlier, we would have been in the army, and it's very doubtful that The Beatles would have formed."
Fellow Beatle Ringo Starr recently claimed that the band were lucky to recruit him as their drummer in the early '60s.
> Paul McCartney to release pop standards album
Watch Paul McCartney perform 'Don't Get Around Much Anymore' below: