U.S. West Texas Intermediate (WTI) crude futures were at $58.24 per barrel at 0541 GMT, down 39 cents, or 0.7%, from their last settlement.
Front-month Brent crude futures, the international benchmark for oil prices, were at $68.58 per barrel, 11 cents, or 0.2%, below their last close.
Both crude contracts last week registered their biggest price declines this year amid concerns that the U.S.-China trade dispute could accelerate an international economic slowdown.
"Sentiment remains fragile and vulnerable to any deterioration in U.S.-China trade frictions," said Jeffrey Halley, senior market analyst at futures brokerage OANDA in Singapore.
Money managers cut their net long U.S. crude futures and options positions in the week to May 21, the U.S. Commodity Futures Trading Commission (CFTC) said on Friday.
"Some signs of low confidence are creeping into positioning data," Bell said.
In oil futures markets, the trade war effect is better seen beyond the spot market.
"The impact from a trade war is a more medium- to long-term issue and December spreads weakened sharply over the last week," said Edward Bell, commodity analyst at Emirates NBD bank.
Beyond financial markets, there are also signs on the ground of a slowdown in growth in oil demand.
Amid the trade conflict between the United States and China, profits for China's industrial firms dropped in April on slowing demand and manufacturing activity, according to data published by the National Bureau of Statistics (NBS) on Monday.
China's automobile sales, a key driver of global oil demand growth, will reach around 28.1 million units this year, unchanged from levels seen in 2018, when the country's auto market contracted for the first time in more than two decades, state news agency Xinhua reported on Sunday.
The outlook for flat car sales may be too optimistic still, as monthly sales have so far declined for 10 consecutive months.
A bright spot for carmakers, although not for the oil industry, is that sales of new energy vehicles are likely to grow by about 27 percent to hit 1.6 million units, from 1.26 units in 2018, the report said.
Bell at Emirates NBD said spot crude prices were prevented from falling further as the market was "pricing in ... lower supplies from OPEC+".
A group of producers led by the Organization of the Petroleum Exporting Countries (OPEC), known as OPEC+, has been withholding supply since the start of the year to tighten the market and prop up prices.
(Reporting by Henning Gloystein; Editing by Joseph Radford and Richard Pullin)
By Henning Gloystein