“Every day that goes by, my concern for my father and Siamak increases,” Mr. Namazi said this week in a telephone interview from Dubai, where he lives. He expressed particular anxiety about his father, who has a heart ailment, saying that an Iranian prison cell was “no place for an aging old man.”
Last month, on the anniversary of the elder Mr. Namazi’s imprisonment, Unicef exhorted the Iranian authorities to release him. “After a lifetime of humanitarian service, he has earned a peaceful retirement,” a Unicef statement said.
Mr. Trump has frequently railed against the nuclear agreement, describing it as a giveaway to Iran. He also vowed as a candidate to bring home Robert Levinson, an American who has been missing in Iran for 10 years.
The Iranian authorities, who have claimed ignorance about Mr. Levinson’s whereabouts, are holding at least four American citizens of Iranian descent and two permanent residents of the United States. No official talks are known to be underway about releasing them.
But Mr. Trump has made clear that he will not pay what he says amounts to ransom, which the Obama administration was accused of doing after the Iranians released five Americans in January 2016, when the nuclear agreement entered into force.
President Barack Obama’s aides described the payment made to Iran, which totaled roughly $1.7 billion, as the coincidental settling of an old, unresolved debt, an explanation widely regarded by critics — and even some administration supporters — with cynicism.
Their suspicions were reinforced when Iranian Revolutionary Guards officers were later quoted in Iran’s state press as having bragged about the payment. The boasting suggested that some powerful Iranians believed a precedent had been set to demand more money for further prisoner releases.
“Whatever Obama administration officials wish to call it, their de facto ransom payment has encouraged Iran to double down on efforts to target U.S. citizens traveling there,” Saeed Ghasseminejad, an associate fellow at the Foundation for Defense of Democracies, a Washington-based group critical of the Iran nuclear deal, wrote in a recent policy brief. The Americans seized since the deal was completed, he wrote, are “testament to that disconcerting trend.”
In a widely quoted Twitter message in October, after Siamak and Baquer Namazi were sentenced to 10 years in prison on unexplained charges of having aided a hostile foreign power — meaning the United States — Mr. Trump denounced Iran over what he called its demand “for a fortune” in prisoner exchanges.
“This doesn’t happen if I’m president,” he stated.
Iran’s use of American prisoners for negotiation leverage has been a recurrent theme in the four-decade history of its enmity with the United States. The prisoners released after the successful nuclear diplomacy were widely seen as pawns in that effort.
It remains unclear precisely why Iran has continued what appears to be an arbitrary pattern of arresting Americans of Iranian descent, with no obvious evidence of wrongdoing. But many hard-line advocates of anti-American views in Iran, including Ayatollah Khamenei, see Iranian-Americans as particularly dangerous spreaders of seditious ideas.
Other Americans known to be held in Iran are Karan Vafadari, 55, a Tehran art gallery owner, arrested in September along with his wife, Afarin Niasari, 44, an Iranian citizen with permanent United States residency status; and Robin Shahini, 46, a rights activist and San Diego State University graduate student, arrested in July while visiting his sick mother.
What charges, if any, Mr. Vafadari and his wife face have not been disclosed. The couple have been described by friends, including foreign diplomats, as prominent in Iran’s artistic and cultural communities.
Mr. Shahini was sentenced in October to 18 years in prison on charges similar to those in the case of the Namazi father and son, and is believed to be on a hunger strike.
Iran also is holding Nizar Zakka, a Lebanese citizen and internet freedom advocate with permanent United States residency status. He was arrested in 2015 and sentenced in September to 10 years in prison for spying. He also is believed to be on a hunger strike.
Mr. Vafadari’s sister, Kateh Vafadari, who lives in the Washington area, said nothing of the arrests for a few months. But on Dec. 1, she wrote to Ayatollah Khamenei, imploring him to intervene, according to the Center for Human Rights in Iran, a New York-based advocacy group. The center quoted the letter as saying Mr. Vafadari and his wife had been subjected to “extortion, property seizure and national security threats.”
Babak Namazi, who called on Mr. Trump to help his family in an op-ed column published last Friday by The Washington Post, also visited the capital last week, meeting with representatives of the National Security Council and State Department. He said afterward, “I think we heard the right things.”
Mr. Namazi and Jared Genser, a human rights lawyer representing the Namazis, who accompanied him, declined to specify what assurances, if any, they had received, or identify whom they had met.
But Mr. Genser said a senior National Security Council official had “indicated that the Trump administration was rapidly developing a broader Iran policy, of which the hostages would be a key part.”
National Security Council and State Department officials declined to comment on the meetings but reiterated that Americans held unjustly abroad must be released.
Part of the challenge is that in Iran’s view, any American of Iranian descent it incarcerates is an Iranian citizen, not entitled to consular privileges given to foreigners.
During the Obama administration, extensive diplomatic contacts were established with Iran, most notably between Secretary of State John Kerry and his Iranian counterpart, Foreign Minister Mohammad Javad Zarif. That relationship was important in the 2016 release of the five Americans — four of them of Iranian descent.
The Trump administration, by contrast, is not known to be cultivating such relations. Mr. Kerry’s successor, Rex W. Tillerson, is still preoccupied with filling key posts.
The continued inclusion of Iran among the six predominantly Muslim nations in Mr. Trump’s revised visa ban has only aggravated matters, according to Iranian-American advocates. Iran, which has described the ban as insulting, has retaliated by prohibiting most American visitors.
“The problem is that no one has a clue about Trump administration policy,” said Hadi Ghaemi, the executive director of the Center for Human Rights in Iran. With the American prisoners in Iran, he said, “there is limbo, really.”