As to why the Justice Department delayed the payment and now appears to be questioning whether it should be paid at all, Mr. Gilbert said: “I have no earthly idea. We’ve racked our brains for what it could be.”

Ms. Connor said in the letter that the department intended to temporarily reappoint Kenneth R. Feinberg, a Washington lawyer and well-known arbitration expert who was the fund’s original special master and who made the original determination that Mr. Hekmati deserved the compensation, to re-evaluate whether he was eligible.

Mr. Feinberg declined to comment and referred inquiries to Peter A. Carr, a Justice Department spokesman. Mr. Carr said in an email that “we are declining to comment at this time.”

An American of Iranian descent, Mr. Hekmati was born in Flagstaff, Ariz., and grew up in Flint, Mich., son of a microbiology professor and an accountant who had emigrated from Iran. He served in the Marines from 2001 to 2005, doing two tours in Iraq.

He got his bachelor's degree in economics at the University of Michigan and was about to start his master’s there in 2011, but decided first to travel to Iran to visit his grandmother, whom he had never met. Iranian officials detained him two days before he was to return home and accused him of espionage, which he has consistently denied.

Mr. Hekmati was among a group of Americans released by the Iranian authorities in a prisoner swap when the now-endangered nuclear agreement between Iran and major powers took effect in January 2016. He had been imprisoned longer than any of the others and at one point had been threatened with execution.

In letters relayed by his family and supporters at home while he was held in Tehran’s Evin Prison, Mr. Hekmati accused the Iranian authorities of using him as a political pawn based on bogus charges. He also accused them of wanton abuse that included whipping his feet, prolonged solitary confinement, denial of medical care and severe malnutrition.