"No preconditions, no. If they want to meet, I'll meet anytime they want, anytime they want", he said.
Iran's foreign ministry spokesman said Trump's offer to negotiate with Tehran contradicted his actions as Washington has imposed sanctions on Iran and put pressure on other countries to avoid business with the Islamic Republic.
It was not clear how seriously Mr Trump was proposing a meeting, which would be the first encounter between U.S. and Iranian leaders since the 1979 Revolution.
Al Jazeera's Zein Basravi, reporting from Tehran, said Iranian officials' lack of trust towards Trump made the issue of any future talks "a non-starter". "If they want to meet, I'll meet", he told reporters.
The escalation in rhetoric also resulted in strong words from Iran's President Rouhani, who last week said, "America should know peace with Iran is the mother of all peace" and "war with Iran is the mother of all wars". "I'll meet", Trump said at a joint press conference with Italy's Prime Minister Giuseppe Conte.
Rouhani said in a meeting with British Ambassador Rob Macaire Tuesday Tehran will not give up its rights to export oil, and called the US withdrawal from the nuclear pact illegal.
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World powers and Iran have been working on an economic package to compensate for USA sanctions that begin taking effect next month.
Earlier this month, Pompeo compared Iran's leaders to a "mafia" and promised unspecified backing for Iranians unhappy with their government.
"They're certainly entitled to do that", spokeswoman Heather Nauert said during a July 31press conference.
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".
The global nuclear agreement forced Iran to curtail its uranium enrichment capacity to prevent it from developing nuclear weapons, and imposed stringent verification processes, in exchange for relief on crippling sanctions.
Iran's currency plumbed new depths on Monday, dropping past 120,000 rials to the dollar, as Iranians brace for August 7 when Washington is due to reimpose a first lot of economic sanctions following Trump's withdrawal from the nuclear deal.
Washington aims to force Tehran to end its nuclear program and its support of militant groups in the Middle East, where Iran is involved in proxy wars from Yemen to Syria.
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The withdrawal is reverberating through Iran's political system, damaging moderates and reformists who championed the diplomacy and saw relief from sanctions as the ticket to economic prosperity.
But Trump's remarks did represent a marked softening of tone from when he lashed out at Rouhani in a tweet a week ago.
The Iranian leadership has previously ruled out one-on-one talks with Trump, following his decision to pull the US out of the deal under which Iran was given relief from sanctions in return for curbs on its nuclear program.
Trump answered, "Never, ever threaten the United States again or you will suffer consequences the likes of which few throughout history have ever suffered before".
The Trump administration may be more stringent. But Iran isn't going to do this, any more, frankly, than North Korea will.
Given Iran's various centers of power and the skepticism about such a meeting on the part of Iranian Supreme Leader Ali Khamenei, the country's spiritual guide and most powerful man, it's very unlikely to happen.
"I do not have the impression that he is actually seeking a war with Iran", Blanc said of Trump.
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