Brent crude futures had dropped 92 cents, or 1.5%, to $60.97 a barrel by 0640 GMT.
U.S. West Texas Intermediate (WTI) crude futures declined 73 cents, or 1.3%, to $54.93 a barrel.
Both crude benchmarks fell last week, with Brent down 2.5% and U.S. crude falling 1%.
Asian equity markets dropped to a six-month low on Monday while gold prices climbed as investors sought safe-haven assets because of the ratcheting up of the trade dispute between China and the United States, the world's two largest economies.
"Crude oil futures experienced significant headwinds as global risk appetites remain feeble over subdued global growth and a sudden escalation in the Sino-U.S. trade dispute," said Benjamin Lu, commodities analyst at Singapore-based brokerage Phillip Futures.
Trump last week said he would impose a 10% tariff on $300 billion of Chinese imports starting on Sept. 1 and said he could raise duties further if China's President Xi Jinping failed to move more quickly towards a trade deal.
The announcement extends U.S. tariffs to nearly all imported Chinese products. China on Friday vowed to fight back against Trump's decision, a move that ended a month-long trade truce.
On Monday, China let the yuan tumble beyond the key 7-per-dollar level for the first time in more than a decade, in a sign Beijing may tolerate further currency weakness because of the trade dispute.
The 1.4% drop in the yuan came after the People's Bank of China (PBOC) set the daily mid-point of the currency's trading band at its weakest level since December 2018.
A lower yuan would raise the cost of China's dollar-denominated oil imports. It is the world's biggest crude oil importer.
Signs of rising oil exports from the United States also pressured prices on Monday. U.S. shipments surged by 260,000 barrels per day (bpd) in June to a monthly record of 3.16 million bpd, U.S. Census Bureau data showed on Friday.
The trade war and rising supply should accelerate the trend of speculators reducing their bullish positions in the WTI futures markets.
Speculators cut bullish wagers on U.S. crude in the week to July 30, while bearish wagers rose to their highest since February, the U.S. Commodity Futures Trading Commission (CFTC) said on Friday
However, speculators increased their bullish positions in Brent futures.
Also in the United States, the weekly oil rig count, an indicator of future production, fell for a fifth week in a row as most independent producers cut spending even though majors were still pushing ahead with investments in new drilling.
Iran's seizure of an Iraqi oil tanker raised some concerns about potential Middle East supply disruptions in the Gulf. Iran's state media reported on Sunday the Iranian Revolutionary Guards seized the ship for smuggling fuel.