Iranian authorities have repeatedly said that Amiri was seized by the CIA as he visited Saudi Arabia last year. US officials have denied those allegations.
Iranian state media reported that Amiri asked for a "quick return" to Iran, and Basit said Iranian authorities were making arrangements to get him out of the United States.
The US state department said on Tuesday that Amiri was "free to go," and that there was no evidence he was mistreated during his time in the US.
Amiri 'handed over'
Iran's semi-official Fars news agency claimed Amiri was handed over to the embassy by US agents, calling it a defeat for "America's intelligence services".
"Since the day that my word was published on the internet, the Americans found themselves losing in the game"
"Because of Iran's media and intelligence activities, the American government had to back down and hand over Amiri to the embassy on Monday night," Fars said.
Iranian state television has broadcast the text of what it claims is an interview with Amiri conducted on Tuesday. But the authenticity of the interview is impossible to confirm.
"After the film of my interview was published in the internet [which lead to] the disgrace of the US government because of [their responsibility in] the kidnapping," the interviewee said.
"They intended to send me back home without much noise in order to cover up the kidnapping through denying the whole case, but they couldn't do that in the end," it said.
US officials reject the allegation that Amiri was kidnapped, and say he "lived freely" in the United States. Hillary Clinton, the US secretary of state, said that Amiri would try to travel back to Iran via a third country.
"He's free to go, he was free to come, those decisions are his to make," Clinton said.
Amiri's fate has been the subject of confusion for weeks. On June 29, Iranian state television aired a video of a man claiming to be Amiri. The man said he had managed to escape from US intelligence agents in the state of Virginia.
"I could be rearrested at any time by US agents ... I am not free and I am not allowed to contact my family. If something happens and I do not return home alive, the US government will be responsible," he said.
"I ask Iranian officials and organisations that defend human rights to raise pressure on the US government for my release and return to my country," the man said, adding he has not "betrayed" Iran.
US officials dismissed the allegations in the Iranian broadcast.
Before that video, two others said to show Amiri appeared on the internet. In the first, broadcast on Iranian TV, a man said he was abducted and was being held in the United States.
He said he was forced to take part in a media interview "to claim that I was an important figure in Iran's nuclear programme and that I had sought asylum in America of my own free will".
But in a second video, a man also purporting to be Amiri said he was actually studying in the United States.
US-based ABC news reported in March that Amiri had defected and was working with the CIA.
US officials have rejected these allegations, but Iran claims it has numerous citizens in secret detention in the US, including a former deputy defence minister who disappeared in 2007.
Earlier this month, Iranian authorities said that they had evidence that Amiri had been abductedand had handed it over to the Swiss embassy, which represents US interests in Tehran.