Secret talks have been taking place between British Museum chair George Osborne - who is a former UK finance minister - and Greek Prime Minister Kyriakos Mitsotakis for a year, the newspaper Ta Nea reported in Athens on Saturday.
The talks about the marbles' "possible return" are in an "advanced stage," it said.
The ancient sculptures were taken from the Parthenon temple in Athens in the early 19th century by British diplomat Thomas Bruce, the earl of Elgin.
The British Museum's trustees are free to talk to whom they want, Prime Minister Rishi Sunak's official spokesman told reporters in response.
But he stressed: "We have no plans to change the law, which prevents removing objects from the museum's collections, the British Museum's collections, apart from certain circumstances.
"Our position on that hasn't changed," the spokesman said.
Under the 1963 British Museum Act, which updated previous legislation, the museum can only sell or give away items from its collection under three limited conditions.
They include if the trustees decide that "the object is unfit to be retained in the collections of the Museum and can be disposed of without detriment to the interests of students."
Sunak's spokesman refused to say if the museum might be able to seek a special licence from the government to break up the so-called Elgin Marbles collection.
The British Museum says its entire collection stretches to more than eight million objects, and only about 80,000 of them are on public display at any one time.
They include many items now considered by other countries as loot taken by builders of the British Empire, and the government has long been wary of setting a precedent with the marbles.
In a statement on Saturday, the museum said: "We operate within the law and we're not going to dismantle our great collection, as it tells a unique story of our common humanity".
But it also said it wanted "a new Parthenon partnership with Greece".
Britain insists the marbles were legally taken by Lord Elgin when he directed workers to strip entire friezes from the Parthenon.
Elgin sold the marbles to the government, which in 1817 passed them on to the British Museum. Greece maintains they were stolen and has long campaigned for their return.