The dispute over the so-called Cyrus Cylinder, named after the Persian ruler's 6th century BC conquest of Babylon, is a further sign of deteriorating relations between Tehran and London.
Britain is among Western powers pushing for a fourth round of UN sanctions against Iran over its refusal to halt sensitive work they suspect has military aims, something Iran denies.
In February, Iranian media said Tehran had cut links with the British Museum, after setting a two-month deadline late last year for the museum in London to allow the public display in Iran of the Cyrus Cylinder.
The British Museum, which houses a vast collection of world art and artifacts, said in September that plans to hand over the 2,500-year-old clay cylinder had been delayed due to unspecified "practicalities".
State-owned Press TV, in a report posted on its website on Sunday evening, said Iran had now decided to seek compensation.
"The National Museum of Iran has spent about $300,000 for the exhibition and we will demand our loss to be compensated for by the British Museum," Hamid Baqaie, head of the state Cultural Heritage Organization, was quoted as saying.
Apart from the nuclear row, Iranian officials have also accused London of interfering in Iran's internal affairs following its disputed presidential election last year which plunged the major oil producer into political turmoil.
The Iranian government has warned of a possible downgrading of ties in different fields.
Cyrus is regarded as one of ancient Persia's greatest historical figures, creating one of the world's first empires two centuries before Alexander the Great conquered the region.
He captured Babylon, in today's Iraq, in 539 B.C. and freed Jews held in captivity there. He is also credited as the author of a decree inscribed on the Cyrus Cylinder, which some have described as the first charter of human rights.
