Brent was up 22 cents, or 0.4%, at US$63.17 a barrel by 0843 GMT. West Texas Intermediate (WTI) U.S. crude rose 12 cents, or 0.2%, to US$59.44 a barrel.
The prices have remained rangebound in the last three weeks, with Brent between $60 and $65 per barrel and WTI at $57 to $62.
"Oil prices are entering a consolidation phase after swinging wildly last month," Stephen Brennock of oil broker PVM.
"While there are still plenty of reasons to be bullish, market players have become more cautious as infections have surged in Europe, India and some emerging markets, while vaccine rollouts have proved slower than anticipated," he added.
India now accounts for one in every six daily infections worldwide, and other parts of Asia are seeing infection rates rise.
Asian oil demand remained weak and some buyers asked for lower volumes in May partly because of refinery maintenance and higher prices.
The United States has fully vaccinated more than 70 million people but U.S. gasoline demand has not picked up as much as expected.
The U.S. economy is at an "inflection point" amid expectations that growth and hiring will accelerate in the months ahead, but faces the risk of reopening too quickly and sparking a resurgence in coronavirus cases, Federal Reserve Chair Jerome Powell said in an interview broadcast on Sunday.
"There really are risks out there. And the principal one just is that we will reopen too quickly, people will too quickly return to their old practices, and we'll see another spike in cases," Powell said in a CBS interview, recorded on Wednesday.
On the production side, no new oil drilling rigs were started in the United States in the most recent week, a report published by Baker Hughes showed.- Reuters