The request for money, made to the cardinal in charge of St Peter`s Basilica, was refused, Vatican spokesman Federico Lombardi said.
The letter was among several that went missing from Vatican archives in 1997.
Il Messaggero newspaper said a former Vatican employee had demanded a ransom of €100,000 (£72,000) for the letter.
It said the letter had been written entirely in Michelangelo`s hand and signed by him.
This was rare, as the artist normally dictated letters to assistants, writing only his signature, Il Messaggero reported. Other details of the documents have not been disclosed.
It is not clear why the Vatican had not previously made public the theft, which was reported for the first time by the newspaper.
Police are now investigating the case.
The documents were stolen from the archive of the department that is responsible for the upkeep of St Peter`s.
Michelangelo, who lived from 1475 to 1564, helped design the basilica, which was only completed in 1626.
He also painted the ceiling and the altar wall of the adjoining Sistine Chapel.
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