The long-serving DJ, who spent almost three decades at BBC Radio 2, died "peacefully at home" on Monday afternoon with his wife Alicia by his side, a family spokesman said.
Sir Jimmy was one of the original Radio 1 DJs when the station launched in 1967.
He moved to Radio 2 in 1973 and filled the early afternoon slot until he retired in December 2002.
Before becoming a presenter on the airwaves, Sir Jimmy had a number of hit singles during the 1950s, including a cover of the Nat King Cole song, Too Young.
With the loss of Terry Wogan earlier this year we have certainly lost two radio greats.
Very sad.
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