"They had an all-day negotiation," Trump told Fox News.
"We are having very good discussions," he added.
Qatar said on Tuesday that mediators were making progress in efforts to end the Iran war although Iran has denied Trump's assertion that talks were under way.
The Strait of Hormuz "is going to be open very soon," Trump said, warning that if that did not happen, Iran would be hit hard.
"If they back out again, they are going to get hit really hard," Trump said.
Iran blocked the Strait of Hormuz, a main route for energy supplies from the Middle East, after the start of the war.
The Iran war began on February 28 when the US and Israel attacked Iran and Iran responded with its own strikes on Israel and Gulf countries that host U.S. bases.
US-Israeli strikes on Iran and Israeli attacks on Lebanon have since killed thousands and displaced millions.
Trump was asked by reporters travelling with him in California about a report on the Axios news site that an announcement on the Strait of Hormuz could be made on Wednesday.
"It could happen. Tomorrow or the next day," he said, speaking on Tuesday evening California time.
"A lot of progress has been made."
Oil prices initially eased on hopes of a deal that would allow shipping to resume but later inched upward.
Brent crude, the international standard, was about $US80 per barrel on Wednesday - still well below the levels it hit at the height of the conflict.
The US and Iran reached a deal to open the strait in June, only for attacks to resume.
In recent days, Trump has again alternately threatened massive strikes and voiced support for diplomatic efforts.
Iranian and Omani negotiators have finalised a draft deal to reopen the Strait of Hormuz and are awaiting a final approval from Iran's supreme leader, two regional officials told the Associated Press on Wednesday.
The officials, who were briefed on the negotiations, portrayed the potential deal as a temporary solution to the US-Iranian dispute over vessels' transit through the Strait of Hormuz.
They said it is tied to the agreement reached between the US and Iran in June that aimed at ending the fighting and reopening the Strait of Hormuz but which ultimately collapsed.
The officials, who spoke on condition of anonymity to discuss the matter, said the potential deal will pave the way for the US and Iran to resume negotiations to reach a final deal on Iran's nuclear program.
with AP
