According to four sources familiar with the matter, the agreement would extend the truce for another 60 days and allow traffic to flow through the strategic waterway while negotiators tackle difficult issues such as Iran's nuclear program.
If approved by leadership in Washington and Tehran, it would amount to the biggest step towards peace since the conflict began on February 28. News of the possible agreement came after a round of tit-for-tat attacks between the two countries, the latest such incident since the ceasefire took effect in early April.
Trump has not yet approved the deal, the sources told Reuters on Thursday. Iran has yet to comment on news of the proposed deal, which was first reported by Axios. Iran's Tasnim news agency, citing a source close to the negotiating team, said the text of the agreement had not been finalised or confirmed.
"We're not there yet, but we're very close and we're going to keep on working at it," US Vice President JD Vance told reporters in Washington.
"I can't guarantee that we're going to get there, but right now I feel pretty good about it," Vance said.
The Trump administration has several times said a deal to end the fighting was close, only to have Iran dispute or downplay the claims.
The deal would specify unrestricted shipping through the strait and would require the US also lift its blockade of Iranian ports. The US would also lift some sanctions on Iranian oil sales.
Earlier, US Central Command said its forces had shot down five Iranian attack drones and struck a ground control station in the port city of Bandar Abbas that was about to launch a sixth. Kuwaiti forces then intercepted a ballistic missile fired towards the country, which hosts a large US base.
A US official also said no American aircraft were shot down near Bushehr, Iran, contradicting a report by Iran's state television that a US aircraft had been downed there.
The incidents, while limited, highlighted the fragility of negotiations to turn the tenuous ceasefire into a lasting agreement to end the three-month-old war, which has killed thousands and upended global energy markets.
Kuwait condemned the attack and demanded that Iran immediately halt what it called a serious escalation. The violence, the second flare-up this week, coincided with the Muslim holiday of Eid al-Adha celebrated across the region, where multiple countries have been caught up in the conflict.
Trump has repeatedly said an end to the war is close since mid-March, though the two sides have shown little public movement toward common ground.
Iran has called for sanctions to be lifted, foreign assets to be unfrozen, and US forces to be withdrawn from the region. Washington has called for Iran to dismantle its nuclear program, which Tehran says is for peaceful purposes.
Iran says any peace deal must also end US ally Israel's attacks in Lebanon, but that conflict shows no signs of flagging. Israel said it had targeted infrastructure of Iran-backed Hezbollah militants in the southern city of Tyre and had carried out a strike in the capital Beirut.
Israel has displaced hundreds of thousands of people with a push deep into Lebanon in pursuit of Hezbollah. The Lebanese army said a strike had killed one of its soldiers.
The US warned Oman not to get involved in any effort with Iran to impose a toll in the Strait of Hormuz, and Trump on Wednesday threatened to bomb the country, despite a history of economic and military ties between the two countries.
