Brent crude futures for August rose 44 cents, or 0.6%, to $69.16 a barrel by 0424 GMT after settling at their highest in two years on Friday. U.S. West Texas Intermediate crude for July was at $66.78 a barrel, up 46 cents, or 0.7%.
Both contracts are on track for a second monthly gain as analysts expect oil demand growth to outstrip supply despite the possible return of Iranian crude and condensate exports.
Iran has been in talks with world powers since April working on steps that Tehran and Washington must take on sanctions and nuclear activities to return to full compliance with the 2015 nuclear pact.
"We see demand outstripping supply in the order of 650,000 barrels per day and 950,000 bpd in Q3 and Q4 respectively," ANZ analysts said, adding that this includes 500,000 bpd of increase in Iranian output.
The Organization of the Petroleum Exporting Countries and their allies including Russia will meet on Tuesday.
The group known as OPEC+ is expected to stay the course on its plans to gradually ease supply cuts until July.
"We don't think Iran derails any arrangement with OPEC+, hence the already agreed unwinding of cuts from July are expected to continue and are already baked into balances," Energy Aspects analyst Virendra Chauhan said.
Separately, crude output in the United States soared 14.3% in March, the Energy Information Administration reported on Friday, while Baker Hughes data showed oil and gas rigs rising for a 10th month in a row last week.
EIA's data also showed that a cargo of 1.033 million barrels of Iranian crude oil was recorded as landing on U.S. shores in March, the first shipment of Iranian oil to the United States since 1991.