Brent crude futures rose 9 cents to settle at $75.38 a barrel, a 0.12 percent gain. It touched a low of $73.81, its lowest since May 8.
U.S. West Texas Intermediate (WTI) crude futures rose 77 cents to settle at $65.52 a barrel, a 1.2 percent gain. Earlier, WTI hit a session low of $64.22, the lowest since April 10.
The premium for Brent to WTI hit a session low of $9.38, recovering slightly from last week when the spread reached $11.57, the widest since March 2015. That divergence was "overcooked," prompting profit-taking, said Jim Ritterbusch, president of Ritterbusch and Associates.
The U.S. government has unofficially asked Saudi Arabia and some other OPEC producers to raise oil output, three OPEC and industry sources said, although it has not requested a specific figure.
Earlier on Tuesday, Bloomberg reported the U.S. government had asked the producers to increase oil production by about 1 million barrels per day (bpd).
OPEC supply tends to more directly influence Brent, whereas U.S. crude futures are more closely tied to U.S. supply.
The request comes after U.S. retail gasoline prices surged to the highest in more than three years and President Donald Trump in April complained about OPEC policy and rising oil prices. The national average on Tuesday was $2.94 a regular gallon, according to AAA.
It also follows Washington's decision to reimpose sanctions on Iran's crude exports, which could disrupt global oil supply. Iran's crude oil output could fall 1 million bpd as a result of renewed U.S. sanctions, according to a note from Standard Chartered.
"Markets are forward-looking. The fact that Russia, Saudi Arabia and OPEC more broadly have started discussing raising output levels, you've got this pretty swift correction," said Tyler Richey, co-editor of the Sevens Report in Jupiter, Florida.
Saudi Arabia and Russia were already discussing raising OPEC and non-OPEC oil output by around 1 million bpd, sources familiar with the matter said on May 25.
Saudi Aramco has raised its July price for its Arab Light grade for Asian customers by 20 cents a barrel versus June to a premium of $2.10 a barrel to the Oman/Dubai average, it said on Tuesday.
The Organization of the Petroleum Exporting Countries meets in Vienna on June 22 to decide whether the group and non-OPEC producers, including Russia, should raise output to make up for any supply shortfall from Iran and Venezuela.
U.S. crude inventories fell by 2 million barrels in the week to June 1 to 432.8 million, industry group the American Petroleum Institute said on Tuesday, compared with analysts' expectations for a decrease of 1.8 million barrels.
Crude stocks at the Cushing, Oklahoma, delivery hub fell by 1 million barrels, API said.
(Additional reporting by Amanda Cooper in London, Jane Chung in Seoul and Roslan Khasawneh in Singapore; Editing by Marguerita Choy and Chris Reese)
By Stephanie Kelly