Clarke, who has presented the show since 1994, was diagnosed with cancer in 2005 resulting in the amputation of his left leg. After undergoing further chemotherapy, he returned to presenting duties earlier this year but had recently been receiving more medical treatment.
"I am deeply saddened by this news," said BBC director general Mark Thompson. "Nick was not only a terrific colleague, but also a great friend, whom I worked with over a number of years.
"He was one of the BBC's finest broadcasters and a brilliant political interviewer, who was also a great listener. Nick's interviewing style was penetrating but unfailingly courteous.
"This is a very sad day for the BBC and for Radio 4 listeners. All our thoughts are with Nick's family."