OPEC and its oil-producing allies, known as OPEC+, agreed on June 1 to continue to gradually ease production cuts.
The original plan, agreed on in April, was to return 2.1 million barrels per day to the market between May and July 2021.
The group will meet again on July 1 to devise a plan for future production policy.
Brent crude futures traded at $71.17 a barrel on June 1, up around 2.7%, while West Texas Intermediate crude futures traded at $68.65, the highest number in more than two years.
Oil prices in the U.S. rose more than 2% on June 1 following a Bloomberg report that said OPEC and its allies looked set to increase production in July as expected.
Oil prices have climbed more than 30% so far in 2021.
The alliance announced massive crude production cuts in 2020 in an effort to support prices during the COVID-19 pandemic.