Brent crude was up 17 cents, or 0.25%, at $66.56 a barrel by 1110 GMT. U.S. West Texas Intermediate was 9 cents higher at $60.61 a barrel.
OPEC and Russia will continue their cooperation as long as it is "effective and brings results," Russian energy minister Alexander Novak said in an interview on Monday.
OPEC and allies agreed in November to extend and deepen output curbs in place since 2017. The reduction of output could see as much as 2.1 million barrels per day (bpd) taken off the market, or about 2% of global demand.
Still, OPEC needs to do more to balance out the market on a sustainable basis, Bjornar Tonhaugen, head of oil market research at Rystad Energy, said in a note.
"The OPEC cuts didn't fully solve the problem – instead they offer a light bandage to get through the first quarter of 2020," said Tonhaugen.
Possibly adding to extra supply was a deal signed on Tuesday between Kuwait and Saudi Arabia on the Neutral Zone between the two countries.
he deal aims to end a five-year year dispute between the OPEC members and reopen later next year fields which can produce 0.5 million bpd or 0.5 percent of global supply.
While OPEC has been cutting production, U.S. producers have been filling the gap, pumping ever greater amounts of crude to reach a record high of around 13 million bpd in November.
That has helped swell inventories with U.S. stocks up around 1% this year.
Crude stocks are, however, expected to have fallen by about 1.8 million barrels last week, a second week of declines, according to a preliminary Reuters poll.
The weekly government report on inventories has been delayed by two days due to Christmas. The report is normally released on Wednesday at 10:30 a.m. EST (1530 GMT).