Brent crude futures settled at $66.72 a barrel, climbing 20 cents. US West Texas Intermediate (WTI) crude futures settled at $60.21 a barrel, up 1 cent.
Brent has gained 4% this week while WTI posted a 4.7% rise. Both benchmarks fell last week.
Tropical Storm Barry, which is expected to become a hurricane just before making landfall this weekend, boosted crude futures as oil companies in the Gulf of Mexico sliced production.
Nearly 59%, or 1.1 million barrels per day, of crude oil production in the US-regulated areas of the Gulf of Mexico has been cut because of the storm, the US Bureau of Safety and Environmental Enforcement (BSEE) said.
"The crude oil market is being supported by the Gulf of Mexico production shut-in. ... It is going to look to see if Tropical Storm Barry becomes a major flooding event that impacts the refining sector in Louisiana and impacts gas and diesel," said Andy Lipow, president of Lipow Oil Associates in Houston.
The International Energy Agency (IEA) forecast surging US oil output will outpace sluggish global demand and lead to a large inventory build around the world in the next nine months.
"The IEA report is tempering any price rise that we might see from Tropical Storm Barry because the market continues to stumble under the weight of slowing economic growth," Lipow said.
The world energy watchdog's report came a day after the Organization of the Petroleum Exporting Countries predicted a crude glut next year despite an OPEC-led pact to restrain supplies.
The weekly US oil rig count, an indicator of future production, fell for the second straight week, General Electric Co's Baker Hughes energy services firm said. Drillers cut four oil rigs in the week to July 12, reducing the total to 784, the lowest since February 2018.
The market remained on edge as tensions intensified between Iran and the West. Tehran on Friday said Britain was playing a "dangerous game" after last week's seizure of an Iranian tanker on suspicion it was breaking European sanctions by taking oil to Syria.
"Only time will tell whether this turns out to be a case of wishful thinking, but one thing is clear: geopolitical risks are here to stay," said Stephen Brennock, analyst at PVM Oil Associates.