The future of the Elgin Marbles is in the hands of the British Museum, Downing Street has said, amid reports a deal that could see the sculptures returned to Greece is progressing.

No mention was made of the ancient Parthenon sculptures as Sir Keir Starmer met his Greek counterpart Kyriakos Mitsotakis at No 10 on Tuesday morning.

But the Times newspaper reported progress in the negotiations was the reason “why (Mr) Mitsotakis is meeting today with the British prime minister”, citing a Greek official.

Sir Keir’s official spokesman would not be drawn into saying whether the Prime Minister discussed the marbles with Mr Mitsotakis.

“The Government’s position is that we have no plans to change the law that would permit a permanent move and that the case of decisions relating to the care and management and sculptures are a matter for the trustees of the British Museum, which is operationally independent of the Government,” the spokesman told reporters.

Asked if it would be possible to loan the Elgin Marbles to Athens, he said: “Those decisions are entirely for the British Museum.”

A cultural partnership would likely be at the heart of a deal between the museum and the Greek government, which could see the sculptures returned to Athens in exchange for exhibitions of famous artworks.

Discussions about a “Parthenon Partnership” are “ongoing and constructive”, according to the museum.

A spokesperson for the British Museum added: “We believe that this kind of long-term partnership would strike the right balance between sharing our greatest objects with audiences around the world, and maintaining the integrity of the incredible collection we hold at the museum.”

Athens is understood to believe the UK Government would not stand in the way of such a scheme once it is agreed with the British Museum.

Downing Street insisted ahead of the meeting that discussion about the marbles were not on Sir Keir’s agenda.

At the top of their head-to-head Sir Keir said he hoped to “build on our strong bilateral relationship and to talk about our common issues”, while Mr Mitsotakis said Greece saw the UK as “integral in addressing the security challenges that we are facing” in Ukraine and the Middle East.

Tuesday’s visit was the first the Greek leader has made to the UK since a diplomatic row was sparked under the previous government.

Then-prime minister Rishi Sunak cancelled a meeting with Mr Mitsotakis, who had made a push for the sculptures’ return.

Mr Sunak denied having thrown a “hissy fit” over the matter and accused his counterpart of grandstanding.

Mr Mitsotakis had compared splitting the Elgin Marbles from those still in Athens to cutting the Mona Lisa in half.

Greece has long maintained that the Elgin Marbles were illegally removed from their place high atop Athens’ acropolis during a period of foreign occupation.

The marble statues came from friezes on the 2,500-year-old Parthenon temple and have been displayed at the British Museum for more than 200 years.

They were removed by Lord Elgin in the early 19th century when he was British ambassador to the Ottoman Empire.

Some of the remaining temple statues are on display in the purpose-built Acropolis Museum in Athens, and Greece has called for the collections to be reunited.

The Conservatives meanwhile suggested reports of progress in a deal on the Elgin Marbles showed the Prime Minister was ready to “cave into the radical left”.

Shadow culture minister Saqib Bhatti added: “The marbles are protected by an act of Parliament – the PM needs to be clear that he will not allow the law to be changed and block any legal work around that might be devised to allow them to be taken out of this country

“The Prime Minister should be standing up for Britain, our heritage, and our world-class cultural institutions instead of giving in to pressure from campaigners who detest British history.”