The US denied Tehran's foreign minister permission to visit a colleague in a New York hospital unless one of several US citizens being held in Iran is released.
The US State Department said on Saturday that it would only allow Javad Zarif to visit the country's UN ambassador, Majid Takht-Ravanchi, if the Iranian authorities release a detained American citizen.
"Foreign Minister Zarif would like to visit a colleague who is in the hospital receiving world-class care," the State Department said.
"Iran has wrongfully detained several US citizens for years, to the pain of their families and friends who cannot freely visit.
"We have relayed to the Iranian mission that the travel request will be granted if Iran releases a US citizen."
Mr Takht-Ravanchi is being treated for cancer in New York, where Mr Zarif was visiting for the UN General Assembly meeting.
Mr Zarif and other Iranian officials are under strict travel restrictions when they are in the US that limit them to the area around the UN headquarters.
Tension between Washington and Tehran has increased since last year, after US President Donald Trump withdrew from a 2015 deal aimed at curbing Iran's nuclear programme, imposing sanctions on the country.
On Sunday, Mr Zarif denied that his country would interfere with the coming US presidential election .
In an interview on NBC's Meet the Press, Mr Zarif also accused the US of initiating a cyber war with his country and said that "any war the United States starts, it won't be able to finish."
The interview took place in New York.
When moderator Chuck Todd said that US intelligence included Iran among the countries attempting to interfere with the US election, Mr Zarif responded, "We don't have a preference in your election to intervene in that election."
"We don't interfere in the internal affairs of another country," Mr Zarif said later. "But there is a cyber war going on."
The Iranian official cited Stuxnet, a computer virus that is widely believed to be a joint creation of the US and Israel and is blamed for disrupting thousands of Iranian centrifuges in an effort to damage its nuclear programme.
Also on Sunday, Iran's oil minister ordered his country's energy sector to be on high alert to the threat of "physical and cyber" attacks.
Bijan Namdar Zanganeh said on the ministry's website that "it is necessary for all companies and installations of the oil industry to be on full alert against physical and cyber threats".
Mr Zanganeh said precautions were needed due to American sanctions and the "full-scale economic war" that the Islamic republic accuses the US of waging against it.
Washington, Riyadh, Berlin, London and Paris blame Iran for attacks that damaged the Saudi oil sector on September 14 and forced the world's largest crude exporter to sharply reduce production.
