Brothers John and Clarence Anglin joined Frank Morris to tunnel out of their cell 56 years ago and disappeared into the waters of San Francisco Bay.
The men were never seen again and were presumed to have drowned.
But a handwritten letter, sent in 2013 but only made public this week, suggests John Anglin may still be alive and in his 80s.
The US Marshals Service, which is responsible for hunting fugitives and maintains an active file on all three men, said the FBI had examined the letter for fingerprints and DNA and analysed the handwriting but the results were inconclusive.
The letter writer claimed to be John Anglin and said he spent many years after his escape from Alcatraz living in Seattle.
He also said he lived in North Dakota for eight years and currently lives in Southern California.
According to the letter, Frank died in 2008 and John’s brother died three years later.
The writer offers the authorities a deal.
He said: “If you announce on TV that I will be promised to first go to jail for no more than a year and get medical attention, I will write back to let you know exactly where I am. This is no joke.”
The Anglin brothers were sent to Alcatraz after failed escapes from the Atlanta State Penitentiary where they were serving time for robbing banks with their other brother, Alfred Ray Anglin.
They made their escape on June 11, 1962, when they used a drill made from a broken vacuum cleaner motor to widen air vents in their cells and managed to squeeze through a network of pipes to get to the roof.
After fashioning a makeshift raft out of rain coats and inflating life vests they set off for the mainland.
Their disappearance was only discovered the morning after when guards found dummies in the men’s beds.
FBI chiefs maintained the brothers drowned in their escape but the Anglin family insist they survived, stayed in contact with some of them and possibly settled in Brazil.
Their nephew Ken Widner said: “I think Alcatraz was a life change for them. I mean this prison was the last stop for any prisoner.
“After they left, it was a never-come-back trip. They knew if they were caught that it would be the end of the road for them.
“So, in some ways Alcatraz saved them and put them on a new road of life.”
The escape inspired the 1979 film Escape from Alcatraz starring Clint Eastwood as Frank, Jack Thibeau as Clarence and Fred Ward as John.