McCartney played bass guitar for the record and Starr played drums. It is not yet clear whether the musicians can also be heard together in a song. The album is said to be almost ready, but nothing has been officially announced yet.
It would be the first Stones album with original work since 2005. Keith Richards revealed last month that new music was on the way.
The Rolling Stones and The Beatles both broke through in the 1960s. The members of the English bands have often encountered each other, but rarely worked together.
In the Stones number We Love You from 1965, McCartney and John Lennon sing backing vocals. Brian Jones played saxophone for the Beatles song that same year You Know My Name (Look Up the Number). In 1968, Lennon and Yoko Ono competed in the TV special The Rolling Stones’ Rock and Roll Circus.