Demonstrators hold Iran s flag during protest for Mahsa Amini who died in custody of Iran s morality police, in Los Angeles, California, on October 1, 2022. AFP
Namazi, a former UNICEF official, was detained in February 2016 when the 85-year-old went to Iran to press for the release of his son Siamak, who had been arrested in October of the previous year.
The United States has been pressing for the release of these two men and two other Americans amid efforts to revive the 2015 nuclear deal between Iran and major Western powers.
"Wrongfully detained US citizen Baquer Namazi has been permitted to depart Iran, and his son Siamak, also wrongfully detained, has been granted furlough from prison," a State Department spokesperson told AFP.
It added that the older Namazi "was unjustly detained in Iran and then not permitted to leave the county after serving his sentence, despite his repeated requirement for urgent medical attention."
"We understand that the lifting of the travel ban and his son's furlough were related to his medical requirement."
The United Nations said last week that the pair had been allowed to leave Iran, after an appeal from its Secretary-General Antonio Guterres.
Both were convicted of espionage in October 2016 and sentenced to 10 years in prison.
Baquer Namazi was released on medical leave in 2018 and had been serving his sentence under house arrest.
At least two other American citizens are currently held in Iran.
Businessman Emad Sharqi was sentenced last year to 10 years in prison for espionage, and environmentalist Morad Tahbaz, who is also a British national, was arrested in 2018 and released on bail in July.
A drive to salvage the 2015 nuclear deal resumed in late November last year, after talks were suspended in June as Iran elected ultra-conservative President Ebrahim Raisi.
The 2015 deal -- agreed by Iran, the United States, China, Russia, Britain, France and Germany -- offered Tehran sanctions relief in exchange for curbs on its nuclear program.
But the United States unilaterally withdrew in 2018 under president Donald Trump and reimposed biting economic sanctions, prompting Tehran to begin rolling back on its commitments.
On Sunday, the United States rejected Iranian reports that Tehran's release of US citizens would lead to the unfreezing of Iranian funds abroad.
"With the finalization of negotiations between Iran and the United States to release the prisoners of both countries, $7 billion of Iran's blocked resources will be released," the state news agency IRNA said.
But the State Department dismissed any such link as "categorically false."
Billions of dollars in Iranian funds have been frozen in a number of countries -- notably China, South Korea and Japan -- since the US reimposed sanctions.