Brent crude futures reversed early losses and were up 40 cents at $67.41 a barrel by 0852 GMT. Earlier in the session they hit their highest since May 30 at $67.65, after ending Wednesday up 4.4%.
U.S. West Texas Intermediate crude futures were up 33 cents, at $60.76 a barrel, having earlier touched their highest since May 23 at $60.94. They gained 4.5% in the previous session.
A day after Iran warned Britain would face "consequences" over the seizure of an Iranian oil tanker, three Iranian vessels tried to block the passage of a British ship run by BP through the Strait of Hormuz, the British government said. They withdrew after warnings from a British warship.
"What happened was partially expected. We pointed out last week that Iran was likely to do something of the sort," Petromatrix oil analyst Olivier Jakob said.
"They might have created a little bit of disturbance, but nothing came out of it. For now we are in the process of intimidation and psychological warfare.... To have a strong price reaction you need something to really happen."
Oil prices were also supported by a decline in U.S. inventories. U.S. crude stocks fell 9.5 million barrels in the week to July 5, the Energy Information Administration (EIA) said, more than the 3.1 million-barrel draw analysts had expected as refineries ramped up output.
U.S. oil producers on Wednesday also cut nearly a third of their output in the Gulf of Mexico ahead of what could be one of the first major storms of the Atlantic hurricane season.
Fifteen production platforms and four rigs were evacuated in the north central Gulf of Mexico, according to a U.S. regulator, as oil firms moved workers to safety ahead of a storm expected to become a hurricane by Friday.
"There is nothing like an early start to the hurricane season to support oil prices, but looking under the hood of the EIA data, it paints an even rosier picture for U.S. oil markets," said Stephen Innes, managing partner, Vanguard Markets in Bangkok.
"Imports down, exports likely up and refinery utilisation at yearly highs," he said.
Stocks have now fallen for four consecutive weeks, according to the EIA.
U.S. output is rising again after a brief drop from record levels, according to the EIA. Production last week rose to 12.3 million barrels a day.
"Rising U.S. shale production levels, subdued global economic momentum and existing trade uncertainties will cap bullish gains for crude oil futures," said Benjamin Lu, analyst at Phillip Futures in Singapore.