Trump gave few details of the plan to aid ships and their crews that have been "locked up" in the vital waterway and are running low on food and other supplies more than two months since the conflict began.
"We have told these Countries that we will guide their Ships safely out of these restricted Waterways, so that they can freely and ably get on with their business," Trump said in a post on his Truth Social site on Sunday.
The unified command of Iran's armed forces responded on Monday by warning US forces to stay out of the strait.
Its forces would "respond harshly" to any threat, it added, telling commercial ships and oil tankers to refrain from any movement in the absence of co-ordination with Iran's military.
"We have repeatedly said the security of the Strait of Hormuz is in our hands and that the safe passage of vessels needs to be co-ordinated with the armed forces," Ali Abdollahi, the head of the forces' unified command said in the statement.
"We warn that any foreign armed forces, especially the aggressive US army, will be attacked if they intend to approach and enter the Strait of Hormuz."
Earlier, the United Kingdom Maritime Trade Operations agency said a tanker had reported being hit by unknown projectiles in the strait.
The agency said all crew were reported safe in the incident, which occurred 78 nautical miles north of Fujairah, in the United Arab Emirates, but few details were available.
Trump said "Project Freedom" would begin on Monday morning, adding that his representatives are having discussions with Iran that could lead to something "very positive for all".
Hundreds of ships and as many as 20,000 seafarers have been unable to transit the strait during the conflict, according to the International Maritime Organisation.
US Central Command said it would support the effort with 15,000 US military personnel, more than 100 land and sea-based aircraft along with warships and drones.
"Our support for this defensive mission is essential to regional security and the global economy as we also maintain the naval blockade," Admiral Brad Cooper said.
Iran's state-run IRNA news agency called Trump's announcement part of his "delirium", and Ebrahim Azizi, head of the national security commission of Iran's parliament, said on social media platform X that any interference in the strait would be seen as a ceasefire violation.
Iran has been blocking nearly all shipping from the Gulf apart from its own for more than two months, sending energy prices soaring.
Some vessels attempting to transit the Strait have reported being fired on, and Iran seized several other ships.
In April, the US imposed its own blockade of ships from Iranian ports.
Trump threatened that any interference with the new US operation would "have to be dealt with forcefully".
Iran said on Sunday it had received a US response to its latest offer for peace talks a day after Trump said he would probably reject the Iranian proposal because "they have not paid a big enough price".
Trump said on Sunday evening that talks were going "very well" without elaborating.
Iranian state media said Tehran's 14-point proposal includes withdrawing US forces from nearby areas, lifting the blockade, releasing frozen assets, paying compensation, lifting sanctions, ending the war on all fronts including Lebanon, and creating a new control mechanism for the strait.
With AP and DPA
