Brent crude futures were up 21 cents at $66.97 a barrel by 2:29 p.m. ET, off a session low of $65.27. U.S. futures ended Monday's session 68 cents higher at $56.76, after dropping as far as $55.08, near last week's one-year low.
Traders said futures pared losses when energy information provider Genscape reported that crude inventories fell in the latest week.
EU foreign ministers endorsed a French government decision to sanction Iranian nationals accused of a bomb plot in France, three diplomats said. The United States has granted waivers to some of Iran's oil customers.
OPEC, led by Saudi Arabia, is pushing for the group and its partners to reduce output by 1 million to 1.4 million barrels per day to prevent a build-up of unused fuel.
"Those Novak comments seemed to really catch the market's gaze all of a sudden," said John Kilduff, founding partner at energy hedge fund Again Capital.
"The Russians have been consistent on this. They're not as sure as the Saudis are that we're in oversupply."
Novak said Russia planned to sign a partnership agreement, and that details would be discussed at OPEC's Dec. 6 meeting in Vienna.
"For a cut to be successful in supporting the market, they're going to have to present a front that is not fractured and the chance of that is looking less and less likely as Dec. 6 approaches," said Bob Yawger, director of energy futures at Mizuho in New York.
While a large cut would support crude futures, clear signals from producers are needed, Yawger said. "We lack any certainty other than that the market is oversupplied in the U.S. and everybody else is trying to deal with it."
U.S. crude stockpiles have grown for eight straight weeks, and data last week showed inventories swelled by the most in more than a year.
"Oil prices rose (last week) on hope OPEC and partners, will act to reverse bearish sentiment, but from a technical set up, bear mode remains intact," OANDA strategist Stephen Innes said.
A trade dispute between the United States and China has made investors warier about the outlook for oil demand growth.
This month, fund managers cut their bullish exposure to crude futures and options to the lowest since around mid-2017.
Weekly exchange data shows money managers hold a combined net long position equivalent to around 364 million barrels of U.S. and Brent crude futures and options, down from over 800 million barrels two months ago.
"The main trend remains bearish as investors no longer believe in a risk of supply tightness for crude," ActivTrades chief analyst Carlo Alberto De Casa said.
— CNBC's Tom DiChristopher contributed to this report.