The US has denied a claim made by Iran's state media that two of the country's missiles have hit a US warship near Jask Island.
The unconfirmed strike is said to have stopped the US warship from entering the Strait of Hormuz.
The vessel then turned back from the area, Iran's state media said.
Iran's navy has also said it prevented US warships from entering the Strait of Hormuz.
Senior US officials quickly denied any of their fleet was hit by a missile.
A Centcom statement reads: "No US Navy ships have been struck.
"US forces are supporting Project Freedom and enforcing the naval blockade on Iranian ports."
Donald Trump revealed the US Project Freedom operation on Sunday night.
Two missiles hit a US warship near Jask Island after it ignored Iran's warnings, Iran's Fars News Agency reports
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He described the naval operation, which will guide ships "safely" out of the key waterway as a "humanitarian gesture" amid reports many of the vessels were running low on food.
However, Iran argues this is considered "a violation of the ceasefire" as it vowed to protect the strait "with full strength".
Under the operation, warships, drones, 15,000 troops, guided-missile destroyers and more than 100 land and carrier-based aircraft will help guide vessels through the waterway.
Mr Trump wrote on Truth Social: "For the good of Iran, the Middle East, and the United States, 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."
Senior Iranian lawmaker Ebrahim Azizi said in response that any "interference" in the "new maritime regime of the Strait of Hormuz" would be considered a violation of the ceasefire.
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President Trump announced the launch of 'Project Freedom' on Sunday night
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President said representatives from the US were currently engaged in "very positive" discussions, adding that discussions "could lead to something very possible".
He said: "I have told my Representatives to inform them that we will use best efforts to get their Ships and Crews safely out of the Strait.
"In all cases, they said they will not be returning until the area becomes safe for navigation, and everything else.
"This process, Project Freedom, will begin Monday morning, Middle East time."
MAPPED: Where is the Strait of Hormuz? | GB NEWSAli Abdollahi Aliabadi, a senior Iranian military commander, told Fars News: "We maintain and manage the security of the Strait of Hormuz with all our might, and we call on all commercial ships and oil tankers to refrain from taking any action to travel without the coordination of the armed forces stationed in the Strait of Hormuz, so as not to endanger their security.
"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."
The Strait has been closed since March 2 following the outbreak of the conflict between the US, Israel, and Iran.