Brent crude oil futures were at $71.39 per barrel at 0832 GMT, up 56 cents from their last close and heading for a weekly gain of 1.5 percent, their third weekly gain in a row.
U.S. West Texas Intermediate (WTI) crude futures were at $64.19 per barrel, up 61 cents from their previous settlement, set for a weekly rise of 1.7 percent, their sixth straight week of gains.
"We expect oil price to eventually move higher in Q2 as OPEC+ potentially runs the risk of over-tightening the market by maintaining its current course of action," Harry Tchilinguirian, strategist at BNP Paribas, told the Reuters Global Oil forum.
Oil markets have been risen by more than a third this year by supply cuts led by the Organization of the Petroleum Exporting Countries (OPEC), U.S. sanctions on oil exporters Iran and Venezuela, and an escalating conflict in Libya.
The head of Libya's National Oil Corporation warned on Friday that renewed fighting could wipe out crude production in the country.
"We see Brent and WTI prices averaging $75 per barrel and $67 per barrel respectively through the rest of this year, but risk is asymmetrically skewed to the upside," RBC Capital Markets said in a note.
"Geopolitically infused rallies could shoot prices toward or even past the $80 per barrel mark for intermittent periods this summer," the Canadian bank said.
OPEC and its allies will meet in June to decide whether to continue withholding supply, and while OPEC's de-facto leader, Saudi Arabia, is seen to be keen to continue cutting, sources within the group said it may raise output from July if disruptions elsewhere continue.
On the demand side, most of the world's growth in fuel consumption is coming from Asia, where China's economic growth is expected to slow to a near 30-year low of 6.2 percent this year, a Reuters poll showed on Friday.
But concerns over such a slowdown were muted on Friday.
"While macro fears of an economic hard landing may be overblown, the concentration risk of global oil demand (in Asia) remains underappreciated," RBC Capital Markets said.