Trump announced the US would begin guiding ships on Monday through the strategic strait, a major transport route for oil, gas and fertiliser.
US guided-missile destroyers have entered the Gulf as part of the mission to escort ships through the vital Strait of Hormuz waterway, the US military said on Monday.
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The warships "are currently operating in the Arabian Gulf after transiting the Strait of Hormuz in support of Project Freedom," US Central Command said on X, referring to the operation announced by President Donald Trump on Sunday.
Trump said in a social media post on Sunday the US would "guide" ships out of the strait, warning that Iranian efforts to block them "will, unfortunately, have to be dealt with forcefully."
The effort to revive traffic through Hormuz risks unravelling the fragile ceasefire that has held for more than three weeks.
He described part of what he called "Project Freedom" in humanitarian terms, designed to aid stranded seafarers, many on oil tankers or cargo ships, have been stuck in the Persian Gulf since the war began.
Iran's state-run IRNA news agency later called Trump’s proposal part of his "delirium," and the Iranian military command said on Monday that ships passing must coordinate with them.
"We warn that any foreign military force, especially the aggressive US military, that intends to approach or enter the Strait of Hormuz will be targeted," Major General Pilot Ali Abdollahi told state broadcaster IRIB.
The Joint Maritime Information Centre said the US has set up an "enhanced security area" near the Oman side of the strait. It urged mariners to coordinate closely with Omani authorities "due to anticipated high traffic volume."
It warned that passing close to usual routes, known as the traffic separation scheme, "should be considered extremely hazardous due the presence of mines that have not been fully surveyed and mitigated."
Meanwhile, Iran's military claimed earlier on Monday that it had prevented US Navy ships from entering the Strait of Hormuz.
"Additional news will be announced later," a brief statement carried by Iran’s state-run Tasnim News Agency said.
Iranian IRGC-affiliated media said that a US Navy frigate had been targeted by two missiles in the strait.
"The frigate, which was sailing on Monday in the Strait of Hormuz, in violation of navigation and maritime safety rules near (the port of) Jask, was targeted by a missile attack after ignoring a warning from the Iranian navy," the Fars news agency said, without citing a source.
The US military denied on Monday that any of its Navy ships had been hit in the Strait of Hormuz.
"No US Navy ships have been struck. US forces are supporting Project Freedom and enforcing the naval blockade on Iranian ports," Central Command, which oversees US forces in the Middle East, said in a post on X.
The United Arab Emirates said that Iran had fired two drones at a tanker affiliated with its state oil company ADNOC in the Strait of Hormuz, condemning the attack.
"Targeting commercial shipping and using the Strait of Hormuz as a tool of economic coercion or blackmail represents acts of piracy by Iran's Revolutionary Guard Corps," the foreign ministry said, adding that no one had been injured.
The UAE called the incident a "flagrant violation" of UN Security Council Resolution 2817.