The US military said black-and-white footage it filmed from a US aircraft showed Iran's Guards on a patrol boat drawing up to the Kokuka Courageous and removing an unexploded limpet mine from its hull.
"It is Iran's modus operandi to use limpet mines and to show the United States that they have opportunities to attack the flow of oil in the Persian Gulf and the Strait of Hormuz - but at the same time they are not attacking directly".
Asked how to he planned to address Tehran and stop any further similar incidents, Trump said: "We're going to see".
The United States accused Iran on May 13 of orchestrating the attacks on the tankers.
The Japanese owner of the Kokuka Courageous, one of two oil tankers targeted near the Strait of Hormuz, said Friday that sailors on board saw "flying objects" just before it was hit, suggesting the vessel wasn't damaged by mines.
Both tankers were loaded with petroleum products, and the Front Altair burned for hours, sending up a column of thick, black smoke.
Abbas Mousavi, spokesman for Iran's foreign ministry, called the latest U.S. accusations "ridiculous, but also very worrying and dangerous", according to the IRNA state news agency.
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Tensions have spiked further since Trump acted last month to force Iran's oil customers to slash their imports to zero or face draconian US financial sanctions.
The new attacks come amid years of proxy warfare between Iran and Gulf allies Saudi Arabia, the UAE, and others.
Already, Iran says it quadrupled its production of low-enriched uranium.
Lee wrote that tanker attacks from Iran now would be "clumsy timing", noting they coincide with Japanese Prime Minister Shinzo Abe's visit to Tehran and the release of USA resident Nizar Zakka, who had been held in Iran since 2015 on spying charges.
"Unilateral US actions - including its economic terrorism on Iran - are exclusively responsible for insecurity and renewed tension in our region", Mr Zarif said. The Mason later fired Tomahawk cruise missiles at radar installations believed to have been used in the attempted attack on the warship, Pentagon officials said at the time. Tehran has denied any involvement in the incident. "There is actually no evidence and this grainy footage from the U.S. is not substantial enough - its not clear where this footage came from and who is visible in it", she said.
The second tanker, the Front Altair, which was set ablaze by a blast, was still languishing at sea, although the fire that had charred the hull had been put out.
Tensions between the two countries have been steadily increasing since the United States withdrew from an global nuclear deal that curbed sanctions on Iran.
Iran has denied being involved in the attack, accusing the US of carrying out an "Iranophobic campaign" against it.
Tehran's foreign minister Mohammad Javad Zarif on Friday fired back at Washington, accusing the US, Israel and Saudi Arabia of a plot to "sabotage diplomacy", and appeared to insinuate that those countries were behind the assaults.