Brent crude futures dropped 38 cents, or 0.6%, to $65.73 a barrel by 0507 GMT, following a 1.1% rise on Friday. U.S. West Texas Intermediate (WTI) crude futures were down 31 cents, or 0.5%, at $61.83 a barrel, after rising 1.2% on Friday.
Both benchmark crudes fell about 1% last week.
"Market sentiment was dented on worries that surging number of COVID-19 cases in some countries, especially in India, will slash fuel demand," Kazuhiko Saito, chief analyst at commodities broker Fujitomi Co.
Prime Minister Narendra Modi urged all citizens to be vaccinated and exercise caution, saying on Sunday the "storm" of infections had shaken India, as the country set a new global record for the most COVID-19 infections in a day.
In Japan, the world's fourth-largest oil buyer, a third state of emergency in Tokyo, Osaka and two other prefectures began on Sunday, affecting nearly a quarter of the population as the country attempts to combat a surge in cases.
"Investors, including speculators, have been shifting funds from oil markets to grain markets recently as volatility has been much higher in prices of corn and other grains," Fujitomi's Saito said.
Chicago corn, wheat and soybeans hit multi-year highs last week as concerns over cold weather damage to crops across the U.S. grain belt underpinned prices, along with expectations for more use of agricultural products for biofuels.
"There were technical adjustments as the oil markets' rally has been overdone and as the OPEC+ is set to add supply from May," said Naohiro Niimura, a partner at Market Risk Advisory.
"Brent could head down to around $60 a barrel going forward as a recovery in demand will likely be limited without active travel restrictions worldwide," he said.
The Organization of the Petroleum Exporting Countries and allies led by Russia, known as OPEC+, surprised the market at its April 1 meeting by agreeing to ease production curbs by 350,000 barrels per day (bpd) in May, another 350,000 bpd in June and a further 400,000 bpd or so in July.
The producer group will hold a largely technical meeting this week, with major changes to policy unlikely, Russian Deputy Prime Minister and OPEC+ sources said last week.
A technical committee meeting is set for Monday, where market fundamentals and compliance with production cuts will be discussed.
U.S. energy firms, meanwhile, cut the number of oil rigs operating for the first time since March, as rigs fell by one to 438 last week, according to energy services firm Baker Hughes Co.
