The agreement, if finalised, would be the most significant diplomatic breakthrough yet to end the three-month war, which has killed thousands of people and sent global energy prices sharply higher.
The Fars news agency reported that Iran is likely to approve the agreement although it has yet to give a formal response.
"We just made a great settlement of the war with Iran," Trump told reporters at the White House.
"The Strait will officially open as soon as we sign, which could be soon, very soon, maybe over the weekend in Europe," he said.
Vice President JD Vance could sign for the United States, Trump said.
When asked by a reporter at the White House whether Iran's Supreme Leader Mojtaba Khamenei had approved the deal, Trump said: "I understand the answer is yes".
Trump described the deal as "a very strong memorandum of understanding," adding it was "a little conceptual but it's something that's going to get done".
Trump's announcement came after he called off planned military strikes on Iran, citing progress in talks.
Since mid-March, Trump has repeatedly claimed that a deal with Iran to end the war is close.
The two sides have traded strikes throughout the week, straining a ceasefire announced in April.
Three Iranian sources said on Thursday a political understanding had been reached but some issues remained to be discussed in detail, including a mechanism for the release of tens of billions of dollars of Iranian oil revenues frozen in foreign banks.
The deal would temporarily ease Iran's grip on the Strait of Hormuz and end a US blockade on Iranian ports, the sources said.
Unresolved questions over Iran's nuclear development program and its stockpile of highly enriched uranium would be left for future talks.
It was unclear whether such a deal would satisfy critics within Trump's Republican party who say that any agreement must close Iran's path to developing a nuclear weapon.
Opposition from Iran hawks helped derail a previous effort to secure a deal to open the strait.
Trump said on social media the agreement had been approved by "the highest level" of Iranian leadership, as well as other countries in the region including Israel, Saudi Arabia, Qatar and the United Arab Emirates.
Trump's announcement marked a sharp about-turn from a previous post where he said the US could take Kharg Island, Iran's main oil export terminal.
The latest apparent breakthrough after stalled efforts to end the war came as clashes took the shaky two-month truce to the brink.
Iran's foreign ministry said the US attacks had "effectively rendered the ceasefire … meaningless" but notably stopped short of saying it was abandoning it.
with PA
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