Also known as the Parthenon marbles, the artefacts are a collection of 17 ancient sculptures and part of a frieze that decorated the 2,500-year-old Parthenon temple at the Acropolis in Athens.
They were removed from Ottoman Greece by Lord Elgin in the early 19th century and then passed on to the British Museum, where they remain one of the most prized exhibits, making up 30 per cent of the Parthenon sculptures of ancient Greece.
Speaking to the BBC, former UK chancellor Mr Osborne said that he was "reasonably optimistic" that the law could be amended to allow the return of the Elgin Marbles, which have been a source of contention between the two governments for more than 200 years.
He said that he envisaged that the marbles could be seen in London and in Athens, which would be a "win-win for Greece and for us".
When asked if that meant loans, he said: "We're talking to the Greek government about that, about a new arrangement ... what I didn't want to do is force the Greeks to accept things that they find impossible, and equally they can't force on us things that we would find impossible."
But Mr Osborne ruled out a scenario where the sculptures could be handed over permanently, saying it would need a change of UK law.
"If we wanted to send all the Elgin Marbles back then that would require an act of parliament, and that would be beyond my authority," he said.
"But what the museum can do is try and form a new relationship with Greece, a new Parthenon partnership if you like, and we're having those constructive talks.
"I'm reasonably optimistic. But you know, my view is it's better to have tried and failed than never to have tried at all."