While Iranian officials' initial reaction to Trump's offer was overwhelmingly negative, Ali Akbar Nategh Nouri, a senior cleric and member of the influential Expediency Council, said Trump's suggestion Monday that he would be willing to meet with Iranian President Hassan Rouhani should not be categorically rejected.
One of the earliest responses was issued by an aide to President Rouhani, who said on Tuesday that the USA would have to return to the internationally-backed nuclear deal and stop new sanctions against Iran if it wants negotiations with Tehran.
Trump and Iranian President Hassan Rouhani have traded hostile warnings via speeches and Twitter. "It was a ridiculous deal", Trump said.
The US is set to start reimposing full sanctions on Iran from August 6 - a move that has already contributed to a major currency crisis with the rial losing two thirds of its value in six months.
President Donald Trump's offer of dialogue with Tehran belies a hardening of US policy that intensifies economic and diplomatic pressure but so far stops short of using his military to more aggressively counter Iran and its proxies. As Trump took questions today during his joint press conference with Italy's Giuseppe Conte, he was asked whether he'd meet with Iran's leaders in a similar manner to his recent summits with Kim Jong Un and Vladimir Putin.
"You threaten us with an action that is "unprecedented" in the world".
In a meeting with Britain's ambassador Tuesday, Rouhani called America's abandonment of the nuclear deal "illegal", declaring that "the ball is in Europe's court now".
In Washington, a senior State Department official said Secretary of State Mike Pompeo had no plans to meet with Zarif if they both participate at an Asian security forum in Singapore later this week.
The White House said that even though the president "is open to dialogue and negotiation", it did not mean the USA would lift sanctions or re-establish diplomatic and commercial relations.
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Even though Trump on Monday said if Rouhani were to meet with him there would be "no preconditions", he also did not walk back from any of those earlier demands.
He added that he had recently had a "great" meeting with Russian President Vladimir Putin and with representatives from North Atlantic Treaty Organisation. Iran's semi-official ISNA news agency quoted political adviser Hamid Aboutalebi as saying that for talks to happen, the USA needs to rejoin the deal.
"After the USA illegal withdrawal from the nuclear deal, the ball is in Europe's court now", Rouhani was quoted as saying by his official website in a meeting with British Ambassador to Tehran Rob Macaire.
Monday, the president told reporters that not only is he willing to meet with Iranian leaders, he's willing to meet them without any preconditions. Hamid Aboutalebi tweeted on Tuesday, "Respecting the Iranian nation's rights, reducing hostilities and returning to the nuclear deal are steps that can be taken to pave the bumpy road of talks between Iran and America".
Mr Trump tweeted in capitals that Iran would "suffer consequences the likes of which few throughout history have ever suffered before" if it threatened the US.
In his typical carrot-and-stick approach, Trump exerts increasing pressure on Iran, while leaving the regime a way out.
The former was dismissed by critics as resulting in little of substance, and the latter saw a number of gaffes, including Mr Trump appearing to say that he trusted Mr Putin's denial of election meddling over the opinion of USA intelligence agencies.
In May, Trump withdrew the United States from the landmark 2015 global agreement created to deny Tehran the ability to build nuclear weapons.
The Majlis deputy speaker said the situation is not right for negotiations "because it will be accompanied by disgrace and humiliation".