The meeting comes amid reports that talks over a deal which would see the statues moved to Athens have recently progressed.
The status of the sculptures has been a source of diplomatic tension between the UK and Greece for decades.
Greece says they were stolen, but the British Museum rejects that and says they were legally obtained. Talks between Greece and the museum have been going on since 2021.
Starmer is thought to be more open to the statues being relocated than his predecessor, providing a deal can be struck between the Greek government and the British Museum.
The Elgin Marbles were crafted in the 5th century BC and were originally displayed in the Parthenon in Athens. They are considered among the most prized antiquities from the Ancient Greek period.
They were removed by British aristocrat Lord Elgin while Athens was part of the Ottoman Empire and were badly damaged en route to London, where they have been displayed since.
A law called the 1963 British Museum Act prevents the removal of objects from the British Museum's collection.
The trustees of the British Museum are exploring the possibility of a special loan arrangement with Greece.
But the Greek government has previously said it will not agree to a loan as this would acknowledge the British Museum's ownership of the sculptures.