Bootle-born Payne was one of only a handful of players to play for both Merseyside clubs, although the majority of his career was spent with Liverpool.
Over an eight-year career with the Reds, starting in 1948, he played 243 games, including the 1950 FA Cup final defeat to Arsenal, and scored 43 goals.
He left Liverpool to join Everton in 1956 but injuries restricted to him to just six matches for the Toffees before he was forced to retire.
Payne was known as the 'Merseyside Matthews' in reference to former Blackpool, Stoke and England winger Stanley Matthews.
A statement on the Liverpool website read: "Everybody at Liverpool Football Club would like to pass on their condolences to Jimmy's family and friends."
His daughter Shirley Morgan told the site: "For the last eight years my dad has been in a home and it was a bit of a struggle but he fought to the bitter end, bless him. Thankfully he's at peace now."