Japan's Kokuka Sangyo, owner of the Kokuka Courageous, said the ship was hit twice over a three-hour period.
A shipping broker said the vessel might have been struck by a magnetic mine. "Kokuka Courageous is adrift without any crew on board," the source said.
The crew of about 21 or 22 people was picked up by the Coastal Ace vessel, Denis Bross of Acta Marine in the Netherlands told Reuters. He said they were handed to a U.S. Navy vessel.
Taiwan's state oil refiner CPC said the Front Altair, owned by Norway's Frontline, was "suspected of being hit by a torpedo" around 0400 GMT carrying a Taiwan-bound cargo of 75,000 tons of petrochemical feedstock naphtha, which Refinitiv Eikon data showed had been picked up from Ruwais in the UAE.
Frontline said its vessel was on fire but afloat, denying a report by the Iranian news agency IRNA that the vessel had sunk.
Front Altair's 23-member crew abandoned ship after the blast and were picked up by the nearby Hyundai Dubai vessel. The crew was then passed to an Iranian rescue boat, Hyundai Merchant Marine said in a statement.
Iran's IRNA reported that Iranian search and rescue teams picked up 44 sailors from the two damaged tankers and took them to the Iranian port of Jask. The numbers in the Iranian media report could not be independently confirmed.
Iran's state television showed what it said was a video of rescued crew members in Jask, showing them sitting on sofa, chatting and watching TV. There was one woman among them.
Thursday's attacks came a day after Yemen's Iran-aligned Houthis fired a missile on an airport in Saudi Arabia, injuring 26 people. The Houthis also claimed an armed drone strike last month on Saudi oil pumping stations.
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