The ships may have been struck by a torpedo, according to some reports.
The latest incident comes after Washington alleged Tehran used mines to attack four oil tankers off the nearby Emirati port of Fujairah last month, a claim denied by Iran.
As many as 44 crew members have been rescued this morning, Iran said, from the two ships were involved the Kokuka Courageous and the Front Altair.
They have been taken to an Iranian port, according to state TV.
“Forty-four sailors from the two foreign oil tankers which had an accident this morning in the Sea of Oman were saved from the water by the (navy) rescue unit of Hormozgan province and transferred to the port of Bandar-e-Jask," IRNA said.
Crude oil prices rose on news of the attacks.
The Kokuka Courageous was carrying methanol and the Front Altair, a petrochemical feedstock.
The first alert was put out by United Kingdom Maritime Trade Operations, a maritime safety group run by the British Navy.
— Liveuamap MiddleEast (@lummideast) June 13, 2019
Images posted on Twitter showed one of the burning tankers following the incident.
The incident happened some 25 miles off the Iranian coast. The U.S. Navy’s Bahrain-based 5th Fleet was also assisting.
The timing of Thursday’s reported attack was especially sensitive as Japanese Prime Minister Shinzo Abe was visiting Iran on a high-stakes diplomacy mission.
On Wednesday, after talks with Iranian President Hassan Rouhani, Mr Abe warned that any “accidental conflict” that could be sparked amid the heightened US-Iran tensions must be avoided.
His message came just hours after Yemen’s Iranian-backed Houthi rebels attacked a Saudi airport, striking the arrivals hall before dawn and wounding 26 people.
Mr Abe met with Iran’s Supreme Leader Ayatollah Ali Khamenei on Thursday, the second and final day of his visit.
Tensions have escalated in the Middle East as Iran appears poised to break the 2015 nuclear deal with world powers, an accord that the Trump administration pulled out of last year.
Washington accuses Iran 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.