The singer was scheduled to appear at the station’s ‘In Concert’ event on Thursday night, but laryngitis will prevent him from performing.
He said in a statement: “I’m devastated that I won’t be able to perform for BBC Radio 2 In Concert on Thursday night, as I’m under doctor’s orders to rest as I have laryngitis.
“I hope to reschedule the concert at a later date, and would like to wish all the Radio 2 listeners a very Merry Christmas.”
The singer had signed up to perform at the BBC concert to celebrate his 60-year career after winning a High Court privacy case against the corporation in July.
The BBC was ordered to pay Sir Cliff £210,000 in damages over their coverage of a police raid at his home following a sexual assault allegation, for which he was never arrested or charged.
They also agreed to cover his £850,000 legal bill, after a High Court judge ruled that the BBC infringed Richard’s privacy rights “without justification”.
Jeff Smith, Head of Music at Radio 2, said the station was “looking forward to welcoming Cliff back very soon”.
During his appearance on ‘Loose Women’, Sir Cliff, 78, vowed never to quit show business.
“The word ‘retire’... I don’t think that’s in my vocabulary,” he said.
“What I plan to do is what I’ve done now, really.
“I’ve slowed, I’ve made it a level area - I don’t tour for five or six months the way I used to.”