The DJ and TV presenter, most famous for his show Jim'll Fix It, was just two days away from his 85th birthday.
Police were called to his address in Roundhay, Leeds at 12.10pm, where they discovered his body. His death is not being treated as suspicious.
Last month, it was reported that Sir Jimmy had been taken to Leeds General Infirmary after suffering a bout of pneumonia. It is as yet unknown whether the illness was the cause of his death.
Friend and fellow DJ Dave Lee Travis paid tribute to Sir Jimmy, saying: "For some unknown reason, with Jim, you couldn't anticipate that he wasn't going to be around anymore - he was such a powerful presence.
"He was one of the first DJs that actually became a personality."
Born in 1926, Sir Jimmy was the youngest of seven children, and worked in the coal mines during the war.
He became a Radio 1 DJ before fronting the original series of Top of the Pops in 1964. In 1975, he landed his own family show, Jim'll Fix It. At its peak, the show attracted 20,000 letters every week from children and adults who wanted the host to grant their wish.
Renowned for his trademark tracksuits, gold chains and cigars, he was also extremely charitable, and is thought to have raised more than £40m for good causes in his lifetime. A long-time friend of members of the royal family, he was knighted by the Queen in 1990.
Former Radio 1 DJ Mike Read said: "He was an exceptional character in a world that's got greyer, more normal, more dull. He was a one off. He was the first to go into halls and clubs and play records. You knew it was Jimmy, he stood out in the crowd. He was pure, pure showbiz."
