According to Reuters, Brent crude futures were up $1.71, or 2.85%, at $61.68 a barrel by 0908 GMT Thursday, having risen as much as 4.45% to $62.64.
US West Texas Intermediate crude futures were up $1.25, or 2.44%, at $52.39 a barrel. WTI earlier rose as much as 3.85% to $53.11.
Both benchmarks are nevertheless headed for a weekly loss.
Reuters, citing four shipping and trade sources, said two tankers — identified as the Marshal Islands-flagged Front Altair and the Panama-flagged Kokuka — had been hit in suspected attacks in the Gulf of Oman.
The crew of the two tankers were rescued by Iran. They have been transferred to Jask, in southern Iran.
The Marshall Islands-flagged tanker was, according to reports, heading from Qatar to Taiwan when a fire broke out on it.
The Panama-flagged ship was en route from a port in Saudi Arabia to Singapore when it was hit by an accident involving a fire.
The details of the incidents remain sketchy.