In Michael's autobiography Moonwalk, he say's he first met Paul McCartney at a party on a boat in 1978 were Paul told Michael he had written a song called Girlfriend for him, and sang the song to Michael. I don't think they kept in close contact after that party. And by coincidence when Michael was recording Off The Wall, Quincy Jones brought Michael the song Girlfriend for Michael to record.
I'm not sure when Michael and Paul became close friends, but Michael did stay at Paul's house in sussex around early 1982, which is the photo used for the single The Girl Is Mine that was released the same year. The friendship between Michael and Paul ended in 1985 when Michael bought The Beatles songs. And as far as I'm aware the last time Michael and Paul met was backstage at one of Paul's concerts in 1989.
Michael has been a Beatles fan since he was a kid. I have a video of the Jackson 5 on American Bandstand from 1970 with Dick Clarke asking Michael as who his favorite artists are apart from Motown acts, and Michael said "The Beatles and Three Dog Night". Infact Jackie, Marlon and Jermaine all said The Beatles, except for Tito.