That puts both Brent and WTI on track for their highest settles since March 11.
Early data from Moderna Inc's COVID-19 vaccine, the first to be tested in the United States, showed that it produced protective antibodies in a small group of healthy volunteers, the company said on Monday.
"Oil is breaking out higher after a potential coronavirus vaccine showed positive results in a Phase 1 trial and on reports that China's crude demand is almost back towards pre-virus levels," said Edward Moya, senior market analyst at OANDA in New York.
Last week, China said its daily crude oil throughput rebounded in April from a 15-month low in March as refiners cranked up operations to meet renewed fuel demand after lockdowns imposed to prevent the spread of the coronavirus outbreak were eased. "Optimism on the demand side of the oil equation has helped prices climb further, with gasoline demand coming back as governments ease confinement measures," said Rystad Energy's senior oil markets analyst Paola Rodriguez Masiu.
The rally in the June WTI contract, which will expire on Tuesday, suggested last month's historic plunge to negative-$40 a barrel would not be repeated.
July WTI was the more actively traded futures contract with volumes in the second-month contract outpacing the front-month for several days now. The July contract was up about 6.6% to $31.48.
Oil prices were also supported by production cuts by the Organization of the Petroleum Exporting Countries and its allies, including Russia, a group known as OPEC+.
OPEC+ has cut its oil exports sharply in the first half of May, companies that track the shipments said, suggesting a strong start in complying with a new production cut agreement.
OPEC+ agreed to cut supply by a record 9.7 million barrels per day (bpd) from May 1. Saudi Arabia, the world's top exporter, announced last week it would cut an additional 1 million barrels per day in June.
Kuwait and Saudi Arabia have agreed to halt oil production from the joint Al-Khafji field for one month, starting from June 1, Kuwait's Al Rai newspaper reported on Saturday.