
Kelly stepped back from his long-term role at Talksport after being diagnosed with throat cancer in March.
Talksport paid tribute to him on Wednesday, calling him a "much-loved friend" and a "broadcasting titan".
"A true legend of the station, Danny had time for everyone and went out of his way to be kind," they said.
"His genuine warmth and incredible passion for broadcasting will be sorely missed by all of us."
They added: "A cherished part of Talksport for so many years, Danny brought extraordinary wit, warmth and knowledge to our airwaves.
"His passion for football, music and life was infectious, and his kindness touched everyone around him."
Kelly also presented BBC radio shows in the 1990s with Danny Baker, who described him as a "great friend, supreme broadcaster and funniest man I ever knew".
Kelly was a lifelong Tottenham fan and the Premier League club said he was "one of our own".
A club statement added: "Our thoughts and love are with his family and friends at this incredibly sad time. Rest in peace, Danny."
'A genius who loved entertaining the nation'
BBC Radio 5 Live's George Cummins was a friend and former colleague of Kelly at Talksport.
"Danny had an extraordinary knowledge of music and football," he said. "His ability to combine the two on air made him an incredible broadcaster.
"His brilliant wit made him an amazing story teller. Colleagues would often remark that his brain was like an encyclopaedia.
"While he was a genius on-air, it was his personality off-air that would shine through. He was always there to give advice and help. He always took an interest in colleagues' lives away from work.
"He was a very positive man. He loved his job. He loved entertaining the nation. I'll always remember a great friend who taught me so much."