Trump said his Middle East envoy Steve Witkoff and close aide and son-in-law Jared Kushner talked to Iranian officials on Sunday and that discussions would continue on Monday.
"We have had very, very strong talks. We'll see where they lead. We have points, major points of agreement, I would say, almost all points of agreement ... we've had very strong talks, Mr Witkoff and Mr Kushner had them," Trump said.
"All I'm saying is, we are in the throes of a real possibility of making a deal," he told reporters before departing Florida for Memphis.
He declined to say who the United States was speaking with in Iran but said it was not Iran's Supreme Leader Ayatollah Mojtaba Khamenei.
Trump said Iran "had some leaders left."
"We're dealing with the man who I believe is the most respected and the leader," Trump said.
Araghchi's ministry said there were "initiatives" to reduce tensions, the Mehr news agency reported, adding that Iran wanted the US, as the party that began the war, to be a direct participant.
Earlier on Monday, Trump backed down on a threat to target Iran's power network, saying he would postpone any strikes on power plants and energy infrastructure.
The reaction from markets was swift and marked.
Brent crude oil futures fell sharply, the US dollar fell against other major currencies, stock markets rallied and government borrowing costs fell back.
Trump's statement on Monday came after Iran threatened to attack Israel's power plants and those supplying US bases across the Gulf region if the US targets Iran's power network.
On Saturday, Trump warned that Iranian power plants would be destroyed if Iran failed to "fully open" the Strait of Hormuz to all shipping within 48 hours.
Trump set a deadline of 7.44 pm EDT on Monday (10.44am on Tuesday AEDT).
Iranian attacks have largely closed the Strait, which carries a fifth of global oil and liquefied natural gas.
The Strait of Hormuz will be "opened very soon" if the talks work, Trump said, adding that it would be "jointly controlled" although he did not give a clear answer on by who exactly.
A reporter for the US news outlet Axios said earlier representatives of Turkey, Egypt and Pakistan had met Witkoff and, separately, Iranian Foreign Minister Abbas Araghchi.
Axios journalist Barak Ravid reported on X that Witkoff and Kushner were negotiating with the Speaker of the Iranian parliament, Mohammad Bagher Ghalibaf.
More than 2000 people have been killed in the war the US and Israel launched on February 28, which has upended markets, driven up fuel costs and accelerated global inflation fears.
A spokesman for UK Prime Minister Keir Starmer on Monday welcomed the latest US statement.
"Any reports of productive talks are welcome," a spokesman said after being asked about Trump's announcement.
"We've always said that swift resolution to the war is in global interests and the Strait of Hormuz specifically needs to be reopened," the spokesman told reporters.
