Brent crude futures jumped $2.74, or 2.91%, to $96.28 a barrel by 2 p.m. ET (1910 GMT). U.S. West Texas Intermediate (WTI) crude futures rose $2.79, or 3,06%, to $93.86 a barrel at 1915 GMT.
Russian forces killed a group of five saboteurs who breached the country's southwest border from Ukraine on Monday, news agencies quoted the military as saying, an accusation that Ukraine called fake news.
French President Emmanuel Macron said earlier on Monday that U.S. President Joe Biden and Russian President Vladimir Putin had agreed in principle to a summit over Ukraine, but the Kremlin said there were no immediate plans.
The Kremlin announced Putin would sign a decree recognizing two breakaway regions in eastern Ukraine as independent. The European Union warned it would consider sanctions if the regions are recognized as independent.
U.S. markets were closed on Monday for the Presidents Day holiday.
"Oil prices are once again marching upwards, as the optimism of a Biden-Putin meeting fades, while OPEC+ is continuing to struggle to hit its quotas which have largely created the severe global energy deficit," said Pratibha Thaker of the Economist Intelligence Unit.
Ministers of Arab oil-producing countries said on Sunday that OPEC+ should stick to its current agreement to add 400,000 barrels per day of oil output each month, rejecting calls to pump more to ease pressure on prices.
Price gains have been limited by the possibility of more than a million bpd of Iranian crude returning to the market.
Iranian foreign ministry spokesperson Saeed Khatibzadeh said "significant progress" had been made in talks to revive Iran's 2015 nuclear agreement on Monday after a senior EU official said on Friday that a deal was "very, very close".
Analysts said the market remained tight and any addition of oil would help, but prices would remain volatile in the near term because Iranian crude was unlikely to return until later this year.
"If a Russian invasion (of Ukraine) takes place, as the U.S. and UK have warned in recent days, Brent futures could spike above $100/bbl, even if an Iranian deal is reached," Commonwealth Bank analyst Vivek Dhar said in a note.