 "Oil edges up on Greek debt hopes, but market still oversupplied")
The euro rebounded against the dollar and equity markets around the globe surged to record highs on Friday after euro zone finance ministers agreed in principle to extend heavily indebted Greece's financial rescue by four months.
That optimism helped lift oil prices from early morning falls.
Benchmark Brent crude futures were trading at $60.34 a barrel by 0532 GMT, up 12 cents from their last settlement, while US WTI crude was up 6 cents at $50.87 a barrel.
Despite the rises, prices remain below last week's highs as the market remains oversupplied.
US oil production growth is expected to reach 440,000 barrels per day by the fourth quarter of 2015 compared with a year before, based on the current rig count, Goldman Sachs said in a note.
US drilling has slowed since the beginning of the year as producers react to a collapse in oil prices that started in June 2014. But Goldman said the the pace of the US rig count decline was also slowing down.
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"The current rig count is pointing to US production growth decelerating close to the level required in our view to balance the oil market," the bank said.
Following the fall from June 2014 to six-year lows, oil prices have picked up since mid-January, with Brent jumping almost $20 a barrel to touch $63 a barrel last week as traders closed long-standing short positions in reaction to the falling U.S. rig count.
Analysts said that a cold spell in the Unites States, which has hit refinery output, could prevent crude prices from rising further.
Major US East Coast refineries have been hit by cold weather, sending up heating oil futures on fears of tight supplies.
"Since the amount of crude oil being refined is less than usual, we would be seeing excess crude oil in the U.S. market ... This causes crude oil to decrease in price while refined products increase in price," Phillip Futures said on Monday.