Trump posted on social media that the emerging agreement would reopen the strait, the vital shipping passage whose closure has sparked a global energy crisis since the US and Israel launched the war on Iran in February. He did not say what else would be included in an agreement.
"Final aspects and details of the Deal are currently being discussed, and will be announced shortly," Trump wrote on Truth Social on Saturday.
Various media in the US and Iran said the memorandum that could yield an agreement lays out a phased framework for ending months of fighting, reopening the waterway soon and lifting a US blockade on Iran.
Plans for Iran's stockpile of enriched uranium, which Washington has insisted it give up, would be negotiated within 30 to 60 days, the reports said.
US Secretary of State Marco Rubio, on a visit to India, said more news could come on Sunday and there was a possibility of good news on the strait within hours.
A senior Iranian source told Reuters that if Iran's Supreme National Security Council approved the memorandum, it would be sent to Supreme Leader Ayatollah Mojtaba Khamenei for final approval.
But Iran's Tasnim news agency said differences remained over one or two clauses. Tasnim cited a source as saying there would be no final understanding if the US continued to create obstacles.
A deal cementing a fragile ceasefire would bring relief to markets but would not immediately quell a global energy crisis, which has driven up costs of fuel, fertiliser and food.
Even if the war ends now, full oil flows through the strait will not return before the first or second quarter of 2027, the head of the United Arab Emirates' state oil firm ADNOC said last week.
Axios reported late on Saturday that the US and Iran were close to a deal, which it said would include no tolls on ships transiting the strait, while Iran would be able to freely sell oil.
In exchange, the US would lift its blockade on Iranian ports and waive some sanctions on Iranian oil, the US news outlet said, citing a US official.
The draft agreement also includes commitments from Iran never to pursue nuclear weapons, Axios said.
Trump, while offering various war aims during the conflict, has repeatedly said the US struck Iran to prevent it from obtaining nuclear weapons.
Iran has long denied it is pursuing such weapons and says it has a right to enrich uranium for civilian purposes, although the purity it has achieved far exceeds that needed for power generation.
Iran's Fars news agency said the draft also stipulates that the US and its allies will not attack Iran or its allies, and in return Iran pledges not to launch pre-emptive attacks on them.
Trump, whose approval ratings have been hit by the war's impact on US energy prices, said on Friday he would not attend his son's wedding this weekend, citing Iran among the reasons for staying in Washington.
Trump spoke on Saturday with leaders from Saudi Arabia, Qatar, the United Arab Emirates, Jordan, Egypt, Turkey and Pakistan, who encouraged Trump to agree to the emerging framework, Axios reported.
