Iran wants any permanent peace deal in the war to allow Tehran to demand fees for ships passing through the strait.
It has not specified how a fee structure would work, but an Iranian official told Reuters before June's interim truce that it would depend on factors including the type and size of ship, its cargo and other unspecified conditions.
Iran regarded efforts by vessels to transit the strait near the Omani coastline during the ceasefire as a violation of the terms of the truce and fired on them.
It says all ships passing the strait must liaise with a new Persian Gulf Strait Authority that it has set up to administer transit.
Senior Iranian officials say that formalizing its control over the strait is now the most important goal for Tehran in its approach to the conflict.
After the ceasefire broke down in July, U.S. President Donald Trump said Washington would charge for use of the Strait of Hormuz, seeking 20% of the value of all cargo shipped.
That declaration, made in a social media post, appeared to contradict his own earlier statements that Hormuz transit should be free, as well as those of other senior figures in his administration.
On June 25, U.S. Secretary of State Marco Rubio said "no country on Earth has the right to charge for the use of international waterways" and that fees for shipping would never be part of any peace deal.
Trump later said that nobody should charge a fee for using the strait, while adding that he thought it unfair the U.S. should have to pay for protecting the waterway.
The strait is made up of the territorial waters of Iran and Oman, with the maritime boundary running along the middle.
The United Nations Convention on the Law of the Sea says states bordering straits cannot demand payment simply for permission to pass through.
However, they can impose limited fees for specific services such as piloting, tugging or port services, though these may not be levied more heavily on vessels from any particular countries.
Neither Iran nor the United States is a signatory to UNCLOS, but it is widely regarded as international maritime law and Hormuz as an international strait.
In 1968, Iran and Oman agreed on a traffic scheme with the International Maritime Organization under which major vessels would use sea lanes along the middle of the strait. Iranian mine-laying during the war has now made such passage unsafe according to the IMO.
No such unilateral move to demand fees to traverse a strait has been made in modern history, shipping industry officials said.
Gulf states, whose main access to the high seas for their vital energy exports lies through the strait, are particularly concerned about fees.
The Panama and Suez canals, which have been dug rather than occurring naturally, are treated differently than straits.
The Turkish Straits — the Bosphorus, Sea of Marmara and the Dardanelles between the Black Sea and Mediterranean — are governed by the 1936 Montreux Convention guaranteeing free passage for merchant vessels in peacetime.
That convention allows Turkey to levy standardised charges to cover the cost of services but does not allow it to impose a general transit fee.
Singapore does not charge a fee to transit the Singapore Strait.
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