Brent crude futures were up 57 cents, or 0.9%, at $67.09 per barrel by 0642 GMT. The international benchmark settled down 0.7% on Thursday after hitting its highest since May 30 at $67.65.
U.S. West Texas Intermediate (WTI) crude futures were up 46 cents, or 0.8%, at $60.66 a barrel. In the previous session, the U.S. benchmark rose to as much as $60.94, its highest since May 23.
Brent prices have climbed 4.4% this week, while WTI prices rose 5.4%. Both the benchmark crudes posted declines last week.
By Thursday, oil companies in the Gulf of Mexico had cut more than 1 million barrels per day (bpd) of output, or 53% of the region's production, due to Tropical Storm Barry which could make landfall Saturday on the Louisiana coast.
The storm was forecast to become a category one hurricane with winds of at least 74 miles per hour (119 km per hour).
"Brent crude oil ... extended its gains as storms in the Gulf of Mexico halted production of oil and U.S. oil inventories continued to recede more than expected," ANZ Bank said in a note.
U.S. crude oil inventories have decreased for four consecutive weeks. Crude stocks fell 9.5 million barrels in the week to July 5, the Energy Information Administration (EIA) said, a drop that was more than triple the 3.1 million-barrel draw expected by analysts.
Kim Kwang-rae, commodity analyst at Samsung Futures in Seoul, said a sharp drop in U.S. crude stocks and geopolitical risks are expected to keep both Brent and WTI at current levels.
"As geopolitical risks involving Iran are likely to persist, that would support WTI to stay above $60 a barrel, while Brent is expected to stay above $65 per barrel but below $70 for the time being," Kim said.
Iran's alleged attempt to block a British-owned tanker heightened tensions in the Middle East in the wake of attacks on tankers and the downing of U.S. drone by Iran in June.
Iran on Friday called on Britain to immediately release an oil tanker that British Royal Marines seized last week on suspicion it was breaking European sanctions by taking oil to Syria, a foreign ministry spokesman told state news agency IRNA.
"While a full-scale military conflict remains the least likely scenario, the strong increases for cost of insurance will make for a most costly transportation of crude and see new routes explored, delaying crude arrivals," said Edward Moya, senior market analyst at PANDA in New York.
But a lower 2020 oil demand outlook from the Organization of the Petroleum Exporting Countries kept price gains in check. OPEC said the world would need 29.27 million bpd of crude from its 14 members in 2020, down 1.34 million bpd this year.