It is among several historic pieces that went missing from the Holy See's archives in 1997.
Italian media is reporting that a former Vatican employee claims to have the letter and will give it back for around £72,000.
The letter is rare because it was written entirely by Michelangelo whilst most of his other letters were dictated and written by an assistant.
Vatican spokesman Fr Federico Lombardi said the demand was sent to a Cardinal but had been refused.
The letter's theft had not been made public at the time and only came to light in the newspaper report.
Police are investigating the ransom demand and the initial theft.
Michelangelo designed part of St Peter's Basilica and painted the ceiling of the Sistine Chapel before dying in 1564.