Brent crude futures were up $0.34 at $63.73 per barrel at 1117 GMT, equivalent to a weekly rise of 2%. They fell 6% last week.
US West Texas Intermediate crude was $0.4 higher at $56.42 a barrel, a weekly gain of 1.3%. It fell 7.5% last week.
Tensions remained high around the Strait of Hormuz, the world's most important oil passageway, as Iran refused to release a British-flagged tanker it seized last week in the Gulf.
Denmark welcomed on Thursday the British government's proposal for a European-led naval mission to ensure safe shipping through the strait and would consider a military maritime contribution.
The United States is separately working on a multinational maritime security initiative in the Gulf.
However, oil prices' reaction to the strains in the Gulf has been relatively muted. "It appears that the majority of market participants do not expect a military conflict that would hamper oil shipments," Commerzbank analyst Carsten Fritsch said.
Prices also drew support from a crude inventory draw in the United States, but gains were limited as the fall appeared to have been largely anticipated. US production in the Gulf of Mexico was still feeling the effects of Hurricane Barry.
"Several indicators pointing to a slowdown of global oil demand growth appear to have taken over market sentiment," Jefferies analyst Jason Gammel said.
Reuters polls taken from July 1-24 showed the growth outlook for nearly 90% of the more than 45 economies surveyed was downgraded or left unchanged. That applied not just to this year but also 2020.
"Growing challenges in the macroeconomic environment have kept bullish bets in check as risk appetites remain soft over potential weakness in global fuel demand," said Benjamin Lu, commodities analyst at Singapore-based Phillip Futures.
The slowdown in global manufacturing and trade, and the associated hit to oil consumption, largely stems from a US-China trade war that has rumbled on over the last year.
Trade talks between the two countries broke down in May after nearing agreement. Next week, top US and Chinese negotiators will meet for the first time since then. Any positive outcome from the meeting is expected to boost oil prices.
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