A spokeswoman for the clinic would not comment on reports that complications have left Sir Paul unconscious. "The family do not want to say any more about this," she said,
"He is currently recovering from the operation," the clinic said. "His family would like to extend their thanks to all those who have expressed their kind concern, which is much appreciated."
Sir Paul,who was born in America, divides his time between his 3,000- acre estate at Wormsley, Buckinghamshire, and a home in central London.
He was awarded an honorary knighthood in 1986 for services to charity. It was converted to the full honour 12 years later after he received British citizenship.
His family's wealth was originally created in the US through Standard Oil, which had a virtual monopoly on the industry early in the century.
His most famous donations include pounds 50m to the National Gallery, pounds 20m to the British Film Institute, pounds 2m for the new Mound Stand at Lord's cricket ground (He is an ardent fan of the game), and pounds 1m to keep Canova's Three Graces statue in Britain.
But he has also donated to numerous other causes, including striking miners' families and cash-strapped students.