Baqaie said the move concerned archaeological, exhibition and research co-operation.
The museum in London, which houses a vast collection of world art and artefacts, said in September that it had planned to hand over the 2,500-year-old clay cylinder, but that "practicalities" had meant delays.
The two countries had agreed to display the cylinder in Iran for a period of four months beginning on January 16, Hassan Mohseni, Iran's heritage organisation spokesman, said.
The British Museum had promised the loan as far back as 2005, following its exhibition, Forgotten Empire: The World of Ancient Persia.
The Bloomberg news agency reported last month that the museum was monitoring the Iranian political situation to make sure the loan was made in the best possible conditions.
|The British Museum had promised the loan in 2005 after its 'Ancient Persia' exhibition [AP]
But the museum authorities said in January that it would be further delaying the return of the Cylinder because of a discovery.
"We wanted to have discussions with our colleagues in Iran in terms of what we should do," Hannah Boulton, head of press and marketing at the British Museum, said.
Inscriptions similar to those on the Cylinder were found on two cuneiform [writing system] tablets from Babylonia in the museum's collection.
The pieces would be studied to shed light on the Cylinder's "missing" or "obscure" passages, the museum said last month, and presented at a London workshop involving Iranian colleagues.
After that, "it is intended that the two new pieces should be exhibited for the first time in Tehran, together with the Cylinder itself", the museum said.
Boulton said: "The agreement has been made with our colleagues in Iran that we'll postpone the loan to investigate this exciting discovery with them.
"That's the reason for the postponement."
|Only a few scholars can translate cuneiform such as that on the Cyrus Cylinder [AP]
Asked why it took so long for the two tablets to be found, Boulton said: "There are 200,000 cuneiform tablets in our collection, and only a limited number of scholars who can understand and translate cuneiform."
Baqaie said his organisation would send a letter of complaint to Unesco, suggesting Iran had incurred financial costs as a result of the museum's delay in lending it the cylinder.
He also said Iran would be sending letters to the world's museums to inform them of the "political motives" of the British museum, the Iranian semi-official Fars news agency reported.
Cyrus is regarded as one of ancient Persia's greatest historical figures, creating one of the world's first empires two centuries before Alexander conquered the region.
He captured Babylon, in today's Iraq, in 539 BC and freed Jews held in captivity there.
He is also credited as the author of a decree inscribed on the cylinder named after him, which some have described as the first charter of human rights.