The incident followed last month's nearby sabotage attacks on vessels off the Fujairah emirate, one of the world's largest bunkering hubs.
Brent crude futures were up $1.91, or 3.18%, at $61.88 a barrel by 1039 GMT, having risen as much as 4.45% to $62.64.
U.S. West Texas Intermediate crude futures were up $1.42, or 2.78%, at $52.56 a barrel. WTI earlier rose as much as 3.85% to $53.11.
"This is a fairly small increase given the uncertainty and the potential knock-on effects of attacks such as these. This partially reflects the fact that the oil market has already priced in the supply and geopolitical risks emanating from Iran," said Cailin Birch, economist at The Economist Intelligence Unit.
"However, it also reflects market concerns that the continued U.S.-China trade war will weigh on economic activity, and therefore oil demand growth, in the world's two largest economies."
Both crude benchmarks are set for their biggest daily rises since early January, but they are nevertheless headed for a weekly loss.
Oil prices had slumped in the previous session on an unexpected rise in U.S. crude stockpiles and a dimming outlook for global oil demand.
The Bahrain-based U.S. Navy Fifth Fleet said it was assisting the tankers after receiving distress calls following "reported attacks". The United Kingdom Maritime Trade Operations, part of the Royal Navy, said it was investigating.
Iranian search and rescue teams have picked up 44 sailors from two damaged tankers in the Gulf of Oman, the Islamic Republic News Agency reported.
U.S. National Security Adviser John Bolton said on May 29 that naval mines "almost certainly from Iran" were used to attack the tankers off the United Arab Emirates last month, and warned Tehran against conducting new operations.
Tensions in the Middle East have escalated since U.S. President Donald Trump withdrew from a 2015 multinational nuclear pact with Iran and reimposed sanctions, notably targeting Tehran's key oil exports.
Iran, which has distanced itself from the previous attacks, has said it would not be cowed by what it called psychological warfare.
Also supporting oil bulls were signs that OPEC members were close to agreeing on continued production cuts.
(Additional reporting by Aaron Sheldrick in Tokyo; Editing by Dale Hudson) ((Shadia.Nasralla@thomsonreuters.com; +44 207 542 5083; +44 778 99 43141; Reuters Messaging: Reuters Messaging: firstname.lastname@example.org))