A BBC spokesperson said on Tuesday that Cooke, who was credited with improving transatlantic understanding for more than half a century, died at his home in New York.
He became a household name in Britain with his weekly programme which for 58 years explained with a unique personal style the often seemingly arcane inner workings of American politics.
Cooke retired from the BBC in March. He said he had decided to quit the show, the world's longest-running speech radio programme, due to ill-health and on advice from his doctors.
In a statement when he left, Cooke said he had thoroughly enjoyed his years on the airwaves and hoped some of the enjoyment had passed over to the listeners "to all of whom I now say thank you for your loyalty and goodbye".
"I seem to be perceived in America as a benign old English gentleman and in England as an enlightened American"
America knew him as the man who explained all things British and Britain as the man who explained all things American.
Cooke maintained a unique status between the two nations "divided by a common language," as George Bernard Shaw once described the relationship between Britain and the US. "I seem to be perceived in America as a benign old English gentleman and in England as an enlightened American," he once said.