Regional media said there were explosions and fires on both vessels, the Front Altair and Kokuka Courageous.
The crew from both vessels have been safely evacuated, their operators say.
The incident comes a month after four tankers were attacked off the United Arab Emirates, which blamed an unnamed "state actor".
The US said Iran was behind the attacks, which involved naval mines, an accusation Tehran denied. The attacks increased long-standing tensions between Iran and the US and its allies in the Gulf.
Oil prices rose as much as 3.9% from a near five-month low following Thursday's incident, Bloomberg reports.
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Josh Frey, of the 5th Fleet, said in a statement: "US naval forces in the region received two separate distress calls at 06:12 local time (08:12 GMT) and a second one at 07:00.
"US Navy ships are in the area and are rendering assistance."
The UK Royal Navy-linked UK Maritime Trade Operations (UKMTO) safety group earlier issued a warning, urging "extreme caution" in the area.
Wu I-fang, a spokesman for Taiwan's state oil refiner CPC Corp, which chartered the Front Altair, said it was carrying 75,000 tonnes of naphtha and was "suspected of being hit by a torpedo", although this has not been confirmed.
He said all the crew had been rescued from the Marshall Islands-flagged vessel.
The operator of the Panama-flagged Kokuka Courageous, BSM Ship Management, said its crew abandoned ship and were rescued by a passing vessel.
The tanker was carrying methanol and was not in danger of sinking, a spokesman said. It is currently located about 80 miles from Fujairah in the UAE and 16 miles from Iran.
The US under President Donald Trump has taken a hard line towards Iran, accusing it of being a destabilising force in the Middle East.
The US sent an aircraft carrier strike group and B-52 bombers to the region at the start of May in response to what it said was an unspecified plan by Iran-backed forces to attack US forces in the area.
Iran has rejected the claims and has accused the US of aggressive behaviour.