West Texas Intermediate crude was up $0.5, or 0.87%, at $57.93 a barrel.
Brent futures were up $0.09, or 0.14%, at $65.29 a barrel by 1040 GMT.
US President Donald Trump said on Friday he called off a military strike in retaliation for the shooting down of a US drone by Iran, saying the potential death toll would be disproportionate, adding on Sunday that he was not seeking war.
Oil prices surged after Iran shot down the aircraft on Thursday that the United States claimed was in international airspace and Tehran said was over its territory.
Brent racked up a gain of about 5% last week, its first weekly gain in five weeks, and WTI jumped about 10%, its biggest weekly percentage gain since December 2016.
But US Secretary of State Mike Pompeo said "significant" sanctions on Iran would be announced on Monday aimed at further choking off resources that Tehran uses to fund its activities in the region.
British Foreign Minister Jeremy Hunt said the UK believed neither the United States nor Iran wanted a conflict but warned tensions could lead to an "accidental war".
Also boosting prices, global supply may remain tight as OPEC and its allies including Russia appear likely to extend their oil cut pact at their meeting July 1-2 in Vienna, analysts said.
"An extension of OPEC+ production cuts through the end of the year seems highly likely given recent price action," US investment bank Jefferies said in a note.
"The market expects an extension though, and any failure could see oil price gap down. The probabilities favour restraint, however," it added.
Russian Energy Minister Alexander Novak on Monday said international cooperation on crude production had helped stabilise oil markets and is more important than ever.
"There is a good example of successful cooperation in balancing the oil market between the OPEC countries and non-OPEC. Thanks to joint efforts, we today see a stabilisation of world oil markets," Novak said.
Boosting oil demand, prospects of a near-term interest rate cut by the Federal Reserve aimed at bolstering the US economy have weakened the dollar.
Oil is usually priced in dollars, and a slide in the value of the weaker greenback makes it cheaper for holders of other currencies.
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