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President Trump has said there have been major points of agreement between the US and Iran in discussions aimed at ending current hostilities.
Earlier, he called off American strikes on Iranian energy infrastructure, but Israel has continued striking what it says are targets in Tehran.
Earlier, Iran fired missiles at Israel, which said it intercepted them, preventing casualties.
Mr Trump claimed Iran had made contact with the United States and had agreed to not pursue nuclear weapons, adding that the US would continue its bombing if talks failed.
Mr Trump said his envoys Jared Kushner and Steve Witkoff had been speaking to a top person in Iran who was not the new supreme leader, Mojtaba Khamenei.
US media is reporting that this is the Speaker of the Iranian Parliament, Mohammad Bagar Ghalibaf.
In Florida, before boarding Air Force One, Mr Trump spoke to the press.
President Trump earlier wrote on his Truth Social website that he'd ordered the US military to pause its planned strikes on Iranian power plants for five days.
He'd threatened to obliterate, his words, Iran's power plants if it didn't end its chokehold on the Strait of Hormuz, the crucial waterway that allows the passage of oil tankers.
Iran had warned that it would respond by attacking power plants in Israel and those supplying US military bases in the Gulf.
Financial markets have reacted positively.
Oil prices fell sharply, with the benchmark Brent crude price dropping by more than 10%.
As we record this podcast, it's now trading at under $100 a barrel.
Our security correspondent Frank Gardner is in Doha, in Qatar, and explained why he thought Mr Trump had changed course.
Frank Gardner.
Iran, though, has denied there is any dialogue with the United States.
A foreign ministry statement accused President Trump of trying to reduce global energy prices to gain time for his military plans.
But it did acknowledge initiatives from regional countries to reduce tensions.
A news agency affiliated to the powerful Revolutionary Guards said Mr Trump had backed down in the face of Iranian threats against power infrastructure in Gulf countries.