Agreement between Iran and Oman over the strait is seen as vital to a wider deal to end the war that began with US and Israeli attacks on Iran on February 28 and led to Tehran establishing a chokehold over the major energy export route.
A US official, who declined to be identified, had said on Friday that Washington anticipated an agreement soon between Iran and Oman, which sit on either side of the strait, so that normal oil traffic could resume.
Iranian Foreign Minister Abbas Araqchi said Iran and Oman were "very close" to agreement on a new shipping route through the strait but that reopening it depended on other conditions, including US compensation to Iran.
The secretary of Iran's top national security body, Mohammad Baqer Zolqadr, listed other demands including ending US threats against Iran, stopping aggression against Iran and its Lebanese, Palestinian, Yemeni and Iraqi allies, lifting a blockade and sanctions on Iran and freeing Iranian assets.
Oman condemned what it described as repeated attacks on vessels while transiting the strait, without assigning blame, and said that negotiations with Iran on arrangements for shipping through Hormuz were "positive and constructive".
"Oman stresses the importance of avoiding any actions that could affect these negotiations and the progress achieved in a manner that takes into account the interests of all parties," the foreign ministry said.
Araqchi said the previous shipping traffic separation scheme through the strait was no longer acceptable to Tehran and that Iran was discussing a temporary route with Oman while technical and legal issues surrounding a permanent route were resolved.
Iran's President Masoud Pezeshkian, a relative moderate, said he believed now was the best time for an agreement.
"There is cohesion, strength, and unity in the country, and as far as I know, Iran is considered victorious and powerful in this war," Iranian news agencies quoted him as saying.
Iran has responded to the US attacks by targeting US bases in Gulf states and Jordan as well as striking shipping in the narrow strait, which carried a fifth of the world's oil and gas shipments before the war.
The UAE said on Saturday that Iran had attacked a carrier affiliated with its state oil company with a missile as it transited the Strait of Hormuz, with no injuries reported, in the latest escalation in attacks on shipping there. There was no immediate comment from Iranian authorities.
In recent weeks, the Trump administration has repeatedly signaled that a deal to open the strait could be close, only to have Iran deny that talks are under way.
It was unclear whether the latest flurry of negotiations would yield a more lasting arrangement.
"Once the deal is announced to restore commercial shipping without impediments, the United States will lift the blockade of Iranian ports," the US official told Reuters on Friday.
US actions would be tied to Iran's implementation of its commitments, he said.
Iran's Revolutionary Guards, who have targeted the ships, also said on Saturday that reopening the Strait of Hormuz depended on Washington accepting Iran's conditions and was not related to the Iran-Oman negotiations.
"Whenever the United States accepts Iran's conditions, the Strait of Hormuz will certainly be reopened," said Guards spokesperson Hossein Mohebbi, according to Iran's Tasnim news agency.
A proposed deal between Iran and Oman to help end five months of war between Iran and the US would give Tehran control over ships entering the Gulf through the strait, a senior Iranian source and two regional officials told Reuters on Wednesday, one of the biggest concessions yet to Iran.
US officials have repeatedly said they would never agree to Iran controlling access to the world's most important trade route for energy supplies.
