According to the Norwegian Maritime Authority, the ship's crew has boarded a passing vessel. The other tanker was adrift without any crew.
US vessels were "rendering assistance" following the "reported attack", the 5th Fleet said in a statement.
The last reported position for both vessels, as tracked by the website VesselFinder, placed them in close proximity with each other and near the Strait of Hormuz, the narrow choke point separating the Gulf of Oman from the Persian Gulf that Iran has repeatedly threatened to close off in response to USA sanctions over Tehran's nuclear weapons program.
Zarif tweeted that "reported attacks on Japan-related" oil tankers in the Gulf of Oman had taken place while Japanese Prime Minister Shinzo Abe was meeting Iran's Supreme Leader Ayatollah Khamenei "for extensive and friendly talks".
The Gulf of Oman lies at the entrance to the Strait of Hormuz, a major strategic waterway through which a fifth of global oil consumption passes from Middle East producers.
There was no immediate confirmation of Thursday's incident from authorities in Oman or the United Arab Emirates, in whose territorial waters the four tankers were hit in May. The United States said Iranian naval mines were "almost certainly" behind the attack.
Iran has denied any involvement.
Saudi Arabia and the UAE said the May attacks on oil assets in the Gulf posed a risk to global oil supplies and regional security.
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On Thursday, two oil tankers operated by shipping companies Frontline and Bernhard Schulte were damaged in a suspected attack near the Strait of Hormuz, the most important oil artery in the world.
"Following two attacks on Member vessels this morning, I am extremely anxious about the safety of our crews going through the Strait of Hormuz", Pao lo d'Amico, chairman of INTERTANKO, said in a statement.
The owner of the Marshal Islands-flagged Front Altair, which was loaded with 75,000 tonnes of flammable oil naphtha, refuted Iranian reports that it had sunk.
Frontline, the Norwegian company that owns the vessel, said it was on fire in the Gulf of Oman.
Reuters cited anonymous sources saying the crew on both tankers had been successfully evacuated and were safe, even though the vessels are reportedly still burning. The manager of the latter said it had been damaged as a result of a "suspected attack" but that its cargo was intact. The injured man is being treated aboard the Coastal Ace vessel.
The Panama-flagged Kokuka Courageous was headed to Singapore from Saudi Arabia with a cargo of methanol, and 21 of its crew jumped and were rescued, according to IRNA.
Iran denies claims that it wants to attack USA forces, with its ambassador to the United Nations telling NBC News in May that the rhetoric coming from Washington was risky and mirrored the run-up to the Iraq War.
"Three explosions on board the vessel were reported".