Evangelos Venizelos asked for the sculptures to be loaned permanently to a specially-built gallery in Athens, for display during the 2004 Olympics.
But the director of the museum, Neil McGregor, said they should stay put.
The marbles are ancient friezes from the Parthenon in Athens and have been at the centre of a tug-of-war between the UK and Greek authorities for years.
Mr Venizelos, on a visit to the museum on Monday, said: "Now is the moment for a Great Britain gesture for the marbles.
"We prepare the Olympic Games of 2004 in Athens, and we must respect this great opportunity."
Mr Venizelos said the issue for Greece was no longer who owned the marbles - about half of which remain at the Parthenon - but their location.
"For us the problem is not the ownership, the historical rights of the British Museum," he said, but "to present the sculpture as a totality."
He added: "I respect very well the global character of the British Museum but now we live in a new era and the globalisation is a very important element also for us."
Pollution
In return for the permanent loan of the sculptures, Athens is offering the rolling loan of other Greek antiquities to the museum.
But Mr McGregor said: "The Parthenon Marbles in the British Museum are in the best possible place for them, and they must remain here if the museum is to continue to achieve its aim, which is to show the world to the world."
On Tuesday Mr Venizelos will visit the culture minister Tessa Jowell to continue his campaign.
The marbles depict the most formal religious ceremonies of ancient Athens - the Panathenaea procession.
Although they once adorned the Parthenon, they were taken to England by the seventh Earl of Elgin and given to the British Museum in 1816.
A recent campaign, Parthenon 2004, backed by more than 90 UK MPs and public figures, called for the marbles to be returned to Athens in time for the Olympic Games.
Groups opposing their return say they have been saved from deterioration from Greek pollution by the museum.