Brent was up 24 cents, or 0.4%, to $69.70 a barrel, while US West Texas Intermediate crude rose 35 cents, or 0.5%, to $67.20 a barrel.
"Boosted by good economic data and risk appetite among investors on the financial markets, Brent is making a renewed bid for the psychologically important $70 per barrel mark," said Commerzbank analyst Eugen Weinberg, Reuters reported.
"Concerns about demand because of the pandemic are giving way to optimism in view of the rapid return of consumers," he added.
Brent and WTI are both on track to post weekly gains of 5% and 6%, respectively.
Analysts expect global oil demand to rebound closer to 100 million barrels per day in the third quarter on summer travel in Europe and the United States following widespread COVID-19 vaccination programs.
"Gasoline demand has now exceeded 2019 levels in many areas," ANZ analysts said in a note.
Robust economic data from the United States, the world's largest economy and oil consumer, also lent support as the number of Americans filing new claims for unemployment benefits fell to the lowest since mid-March 2020, beating estimates.
The prospect of more Iranian oil coming on to the markets also capped the gains.
Iran and global powers have negotiated in Vienna since April to work out steps that Tehran and Washington need to take on sanctions and nuclear activities to return to compliance with Iran's 2015 nuclear pact with world powers.
Balancing expectations of a recovery in demand against a possible increase in Iranian supply, the Organization of the Petroleum Exporting Countries and allies, including Russia, a group known as OPEC+, is likely to stick to the existing pace of gradually easing oil supply curbs at a meeting on Tuesday, OPEC sources said.
"Our balance sheet shows that the market should be able to absorb this additional supply from OPEC+, along with a gradual increase in Iranian output," said ING analyst Warren Patterson.