President Donald Trump, who abruptly called off a two-week US bombing campaign over the weekend in his latest strategic U-turn, said there were "good talks" under way with Iran but threatened to restart strikes unless negotiations deliver.
Washington launched its renewed bombing campaign earlier this month to break Iran's grip on the strait, through which about a fifth of global oil and liquefied natural gas flowed before the conflict.
Iran has said it wants to manage the strait alongside Oman which controls the opposite shore, and charge service fees to ships that use it.
Washington wants to return to the status quo prior to the conflict, when ships were able to pass freely with no payments, and says charging mandatory fees would be illegal.
Under the Omani proposal, Iran would not exercise sole control and fees would be voluntary, a source told Reuters. The system would be analogous to one in place on Asia's Strait of Malacca, where Indonesia, Malaysia and Singapore ask ships to pay voluntary contributions to fund navigation, environmental protection and search-and-rescue operations.
