Bletchley Park was the home of British military intelligence staff during World War II.
Bletchley Park Museum director Christine Large revealed her optimism after speaking directly to one of the mystery men demanding £25,000 for the Enigma encrypter's safe return.
She told BBC News: "I am sure the machine will be back at Bletchley Park soon, for all to enjoy."
The Enigma machine, one of only three remaining, was stolen from the museum in Bucks during an open day in April.
The ransom had been demanded by a Friday midnight deadline but Ms Large said she had been contacted at 0430 BST on Saturday by a man seeking the £25,000.
Officials at the 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.
Ms Large told the BBC: "We had a deal but there was a fear that the machine might be destroyed if the police did not give certain assurances.
"These assurances have been made now and they have gone some way to being accepted."
Ms Large said she did not believe the Enigma machine "was stolen by the people who are in possession of it now".
She added: "There are a lot of things suggesting that we have got someone here who bought in good faith and would like to get out of the heat and return the machine to us but, obviously, doesn't want to be out of pocket at the same time."
Bletchley Park was the home of British military intelligence staff during World War II.
The Enigma team played a crucial part in cracking messages coded by the Nazis on Enigma machines.
Their work in cracking the Enigma codes - used by the Germans to direct operations including U-boat attacks on Allied convoys - proved vital to the outcome of the war.
The stolen Enigma is a rare four-rotor version, one of only three still known to be in existence.
The letters sent to the museum are being treated seriously because a photocopy of the plate showing the machine's identification number - G312 - was enclosed in an earlier letter, while the latest quotes a code word 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.