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French President Emmanuel Macron is expected to announced the proposed loan of the tapestry, which depicts the Norman Conquest of England, during a visit to Britain on Thursday, according to The Times.
The BBC and The Guardian also reported the proposed loan.
Preparations for the relocation are underway following months of discussions between cultural officials in both countries, but it is contingent on tests showing the 11th-century embroidery can be moved safely, The Times said.
However it could be five years before the tapestry — which is nearly 70 metres (230 feet) long and 50 centimetre high — actually arrives on British shores, the paper added.
No decision has been made about where it would be displayed during the loan, it said citing British officials.
The Bayeux tapestry, which dates from around 1077, depicts the story of William the Conqueror and the Battle of Hastings in 1066 and is displayed in Bayeux, in Normandy.
While it has rarely been moved, it was displayed in Paris in 1804, and again briefly at the Louvre in 1945.
President Macron will hold talks with May at a Britain-France summit at Sandhurst, a British military academy, later this week, with co-operation on various issues set to be discussed.
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