Calls to make it easier for British museums to consider returning cultural objects, that could include the Rosetta Stone and Parthenon Marbles, have been rebuffed by ministers.
House of Commons leader Penny Mordaunt told MPs that revisiting legislation was “not a priority”.
Even the British Museum has expressed optimism that a deal may be possible for the return of the 2,400-year-old friezes, while other British museums have acted unilaterally including London's Horniman Museum.
But for now, it appears there will not be accompanying legislative changes to make the process easier.
Lord Vaizey, a Conservative former culture minister, will chair a new body aiming to return the Parthenon Marbles, often called the Elgin Marbles, to Greece.
He has suggested reforming the 1983 National Heritage Act to give greater freedom for museums to dispose of objects in their collection and deal with restitution requests.
Another Tory former culture secretary, Oliver Dowden, drawing on Greece mythology, said Britain risked opening up a Pandora's box and “we will regret it for generations”.
He said: “We are very blessed in this nation to have world-class museums. They are museums of the world, and the world comes to them.
“One of the bulwarks they have against constant claims of restitution is both the British Museum Act 1963 and the National Heritage Act 1983, and I am aware that there will be a debate in the other place about changes to the 1983 Act.”
Mr Dowden asked for MPs to be allowed to debate the legislation so they can express their support for it. He said: “Otherwise, those institutions risk facing a barrage of claims for restitution, some of which may be encouraged more by virtue signalling.
“I can assure you that if we allow this Pandora's box to open, we will regret it for generations to come as we see those artefacts being removed to countries where they may be less safe.”
Ms Mordaunt said: “I can tell (Mr Dowden) that revisiting the National Heritage Act is not a priority for this government.”
Prime Minister Liz Truss has previously ruled out supporting a deal to return the Elgin Marbles to Greece, despite George Osborne, chairman of the British Museum and former Tory chancellor, saying there was a “deal to be done”.
The sculptures — 17 figures and part of a frieze that decorated the 2,500-year-old Parthenon temple at the Acropolis — were taken by Lord Elgin in the early 19th century when he was the British ambassador to the Ottoman Empire, and have been the subject of a long-running dispute over where they should be displayed.
Greek Prime Minister Kyriakos Mitsotakis has called for the marbles to be returned to Greece on many occasions, even offering to loan some of his country's other treasures to the British Museum in exchange.
In a recent interview with the Sunday Times, he said he will raise the subject with Ms Truss on a visit to London this year.
Mr Mitsotakis said: “At a time when Truss will be looking to build her credibility and when the UK is sort of cornered in terms of its overall image after the (queen's) funeral, it will be a fantastic gesture and that's what I'll tell her.”
In 2021, then-prime minister Boris Johnson told the Greek premier during talks at Downing Street that the issue was “one for the trustees of the British Museum”.