Iran criticized the United States on Saturday for what it called a "inhuman" decision to ban Foreign Minister Mohammed Javad Zarif from visiting one of his diplomats at a hospital in New York.
The State Department stated that Zarif is in town for the United Nations General Assembly can only attend the diploma if Iran releases a US citizen.
On Saturday, Iranian Deputy Foreign Minister Abbas Aragqi told the country's official news agency IRNA that the US has taken humanitarian hostage issues for political reasons.
The United States restricted Zarif's movement in New York to just six blocks, preventing him from reaching Tehran's ambassador to the UN, Majid Taht Ravanci, who is undergoing cancer treatment.
"Foreign Minister Zarif would like to visit a colleague who is in the hospital receiving world-class care," a State Department spokesman told NBC News Saturday.
"Iran wrongly detains several US citizens for years to hurt their families and friends who they cannot freely visit. We turn to the Iranian mission that the travel request will be granted if Iran releases a US citizen . "
There are currently at least four Americans in Iran. Human rights groups and UN observers say the detentions are arbitrary and unfounded.
But Iranian President Hassan Rouhani said it was America's turn to release an Iranian citizen after his country liberated Lebanon and U.S. resident Nizar Zaka in June.
President Donald Trump instead limits travel restrictions on Iranians coming to the United States while blaming the country for threatening peace and stability in the Middle East.
On Wednesday, he imposed a visa ban on top Iranian government officials and their family members, blocking them from entering to travel, study or work.
The travel dispute arises after discussions at a meeting of Iranian representatives with Trump at the UN summit failed to materialize. Both sides cited US sanctions on Iran in Iran as a focal point preventing a meeting.
Trump said on Friday that Iran wanted it to lift sanctions as a condition of enforcement. "I said, of course, NO!" he said on Twitter .
Relations between Washington and Tehran have become increasingly strained in recent weeks.
The United States has accused Iran of backing attacks on Saudi Arabia's oil fields two weeks ago, prompting Zarif to promise a "comprehensive war" in the event of a military strike against his country.
In his address to the UN on Wednesday, Iranian President Hassan Rouhani accused the US of "ruthless economic terrorism" for its sanctions, which Trump promised to "significantly increase" amid renewed tensions in the Arabian Peninsula.
Associated Press contributed.