U.S. oil prices rose on Monday, adding to last week's surge, but concerns about the possibility of weakening demand kept their gains in check.
West Texas Intermediate futures climbed 0.8% to settle at $57.90 per barrel.
Last week, U.S. crude surged 10% after Iran shot down a U.S. drone on Thursday in the Gulf, adding to tensions stoked by attacks on oil tankers in the area in May and June that Washington has blamed on Iran.
Iran denies any role in the tanker attacks.
U.S. President Donald Trump said he called off a retaliatory attack on Iran at the last minute after the drone was downed, but Washington has said it will still step up its sanctions on Tehran.
"We have been surprised by the relative lack of crude price response to some of these events until just recently, which probably reflects the prevailing sentiment of a market in oversupply," HSBC said in a note.
"The tightening of supply/demand balances in the coming months is likely to only heighten the market focus on these geopolitical risks unless they abate."
The International Energy Agency (IEA) said this month it had revised down its estimate for crude demand growth in 2019, citing the U.S.-China trade row.
Hopes are waning for progress in Sino-U.S. trade talks at this week's G20 meeting as investors await a meeting between Presidents Donald Trump and Xi Jinping.
"The oil market ... is turning its attention to the upcoming G20 meeting and the (limited) prospect for a deal, and with that, renewed focus on slowing demand growth," said Saxo Bank's head of commodity strategy Ole Hansen.
"Overall, however, the market is looking up instead of down with (the prospect of) OPEC+ agreeing a production cut extension, a weaker dollar and Mideast tension".
Supply still looks to remain relatively tight, as the Organization of the Petroleum Exporting Countries and its allies including Russia, an alliance known as OPEC+, appear likely to extend a deal on curbing output when they meet on July 1-2 in Vienna, analysts said.
Russian Energy Minister Alexander Novak said on Monday that international cooperation on crude production had helped stabilize oil markets and was more important than ever.
But he also voiced concerns about demand, saying extending the deal on supply cuts would depend in part on "the consumption of oil in the third and fourth quarters, (and) the pace of growth of the world economy."