According to his family, the 85-year-old died earlier this week following a stroke, reported BBC online.
White was chosen ahead of Ringo Starr in September 1962 to play drums on the single version of "Love Me Do" and its B-side, "PS I Love You".
White, who was born in Glasgow in 1930, is also believed to have played on the album version of Please Please Me.
He could therefore legitimately claim to be one of the so-called "Fifth Beatles", alongside the likes of Pete Best, Stuart Sutcliffe and others.
He went on to tour with Marlene Dietrich, Burt Bacharach and Rod Stewart and perform with the BBC Scottish Radio Orchestra.
In a 2009 interview with a New Jersey newspaper, White revealed he was often called to London's Abbey Road studios in the 1960s.