Greek Minister of Culture, Elisavet Papazoi, who accompanied the president as he visited the Acropolis in Athens with his daughter Chelsea, said Mr Clinton told her: "'If it would be me, I would give them back immediately."'
She added that the president said he would raise the matter with UK Prime Minister Tony Blair when they meet at a weekend summit in Florence.
The remarks, which US officials emphasised were expressions of Mr Clinton's personal views, were rebuffed by UK Government officials on Saturday.
"There has been no change at all in the government's view that the Elgin Marbles were properly acquired, legally acquired, by the British Museum," a Department of Culture spokesman said.
Theft or purchase?
The classical sculptures, consisting of 17 figures and part of a huge, 160 metre frieze, were taken from the Parthenon temple on the Acropolis in the early 19th Century by Lord Elgin, British ambassador to the Ottoman Empire, which then included Greece.
Lord Elgin later sold the statues to the British Museum in London, their custodians ever since.
Britain has always maintained that its possession of the marbles is legitimate because Lord Elgin paid for them, but Greece says that the deal was illegitimate because they were bought from the Ottoman empire, which was occupying Athens at the time.
Any return of the marbles to Athens would require an Act of Parliament, and the consent of the British Museum.
Chisel damage
Earlier this month, Ms Papazoi said Greek conservationists had discovered evidence of damage caused during a restoration project by the British Museum in the 1930s.
Their report describes the loss of important features during the cleaning project, which employed methods including the use of copper chisels and hard-wire brushes to clean the marble.
The Greek Government wants the marbles returned to a new, as yet unbuilt, museum in Athens in time for the 2004 Olympics to be held in the city.