Iran’s penal authorities granted a four-day leave on Sunday to Baquer Namazi, the oldest of the United States citizens known to be imprisoned in Iran, because of his worsening health, Mr. Namazi’s American lawyer said.

The announcement by the lawyer, Jared Genser, raised speculation that the reprieve was a precursor to parole for Mr. Namazi, 81, a dual Iranian-American citizen who was formerly a diplomat for Unicef. Mr. Namazi’s son Siamak Namazi, 45, also a dual citizen, remains imprisoned.

Parole for the elder Mr. Namazi would be the first serious sign of movement on Iran’s part concerning the Americans known to be incarcerated there since a group was released when the Iranian nuclear agreement took effect two years ago.

The prisoners have long been a sore point in the estranged relations between the United States and Iran, which have worsened under the Trump administration. President Trump has threatened to scrap the nuclear accord, which lifted many economic sanctions on Iran in exchange for verifiable reductions in its nuclear activities.