The strait, through which about a fifth of global oil and liquefied natural gas flowed prior to the conflict, has been effectively blocked since the US-Israel attacks on Iran on February 28, pushing up oil prices and inflation.
Iranian Foreign Ministry spokesman Esmaeil Baghaei said on Monday that talks with Oman on arrangements for navigation through the strait were "progressing smoothly and constructively", with an agreement reached on a shipping route map, while some technical issues on a joint statement remained unresolved.
He said during a weekly press conference in Tehran that the discussions also covered mechanisms for safe navigation, environmental protection, maritime services and combating crime, adding that such services would normally entail receiving payment in return.
On Sunday, Iran's Foreign Minister Abbas Araqchi said an agreement with Oman was in its "final stages" but stressed that Tehran would not reopen the channel unless the US met certain demands made by Tehran.
Iran and the US are not engaged in direct talks, and Tehran will not start them as long as Washington breaches an interim deal signed in June, Araqchi said, adding that messages were being exchanged through intermediaries.
US President Donald Trump said in an interview with Axios that the US was "low-keying" it with Iran. "We are only semi-negotiating with them. We are just watching Iran with its huge inflation and the fact they have no money."
Araqchi said reopening the strait would depend in part on the US paying compensation for damage it caused in widespread attacks on Iran.
The outgoing secretary of Iran's top national security body, Mohammad Baqer Zolqadr, also listed ending further US threats against Iran; halting aggression toward Iran and its Lebanese, Palestinian, Yemeni and Iraqi allies; removing a US naval blockade in the Gulf; lifting sanctions on Iran; and releasing frozen Iranian assets.
Iran on Sunday announced former Revolutionary Guards commander Mohsen Rezaei, seen as a more outspoken figure than Zolqadr, would take over as second-in-command of the body that co-ordinates the country's security and foreign policy.
No reason was given for the change and it was not immediately clear if it would herald any change in policy.
Oil prices edged higher on Monday to $84.22 a barrel, as optimism over a deal to open the strait was tempered by Iran's insistence the US must satisfy its demands.
At the end of April, the global Brent oil benchmark hit about $126 a barrel, the highest since 2022, amid ongoing disruption to shipping.
Trump, who has said the attacks on Iran were aimed at keeping the Islamic Republic from developing nuclear weapons and degrading its ability to threaten the region, is under pressure to end a war that is deeply unpopular at home ahead of midterm elections in November.
Washington and Tehran agreed to a ceasefire in June, but the US reimposed a blockade on Iranian shipping in the Gulf in July, a move Tehran said violated the truce, which by then had already broken down.
Iranian strikes on Strait of Hormuz shipping have been coupled with increased attacks by its Yemen-based allies, the Houthi rebels, who have targeted ships at another oil chokepoint on the other side of the Arabian Peninsula between the Red Sea and Gulf of Aden.
The Houthis also declared a naval blockade against Saudi Arabia in the Red Sea last month in response to what they said was a Saudi siege on them in Yemen, an allegation denied by Riyadh, which backs Yemen's internationally recognised government.
Fighting has escalated between the Houthis and government forces in Yemen, with the Yemeni military on Monday saying seven people including military personnel and civilians were killed in a Houthi attack on the Red Sea port city of Mocha.
Air defences intercepted and shot down 11 Houthi drones that participated in the attack, the military said
