Bletchley Park, in Buckinghamshire, was the home of British military intelligence staff during World War II.
Bletchley Park Museum had agreed to hand over a £25,000 ransom before midnight on Friday but the deadline passed with no word from the mystery man who made the demands.
The Enigma machine, one of only three remaining, was stolen from the Bletchley Park museum during an open day in April.
Officials at Bletchley Park Museum agreed to the handover after deciding an ultimatum delivered in the latest of a string of letters was a genuine threat to the device.
The author of the letters has claimed to be acting on behalf of a third party who unwittingly bought the machine in good faith, only to discover it was stolen.
Thames Valley Police spokesman John Brett said of those behind the ransom: "They are wanting recompense for something they bought in goodwill - they want their money back. We are hoping this person will get in touch."
'Historic vandalism'
Bletchley Park Museum director Christine Large said she was anxious for the machine's safe return: "I very much want it back.
"It would be historic vandalism to destroy this machine. It is an irreplaceable piece of history.
"We have the funds available. We just need to be in direct communication. This is the time to do business."
Bletchley Park, in Buckinghamshire, was the home of British military intelligence staff during World War II.
The Enigma team played a crucial part in cracking Nazi coded messages.
The Nazis encoded their messages on special machines and the stolen one is one of only three remaining.
The claims are being treated seriously because a photocopy of the identification plate of the G312 machine was enclosed in an earlier letter, while the latest quotes a codeword used in earlier demands.
Police want the author of the ransom letters to contact Detective Chief Inspector Simon Chesterman, of Milton Keynes CID, on 01908 686000 and quote reference number 86519900, to negotiate the exchange.