OPEC's ministers will meet in Vienna on Thursday and the wider OPEC+ group will gather on Friday.
It is not certain OPEC+ will agree this week to deepen its curbs.
Beyond that though, the Iraqi oil minister has suggested that perhaps OPEC may even extend production cuts by another 400,000 barrels.
According to Rystad Energy's estimates, the global oil market will be fundamentally oversupplied to the tune of 0.8 million barrels per day (bpd) in the first half of 2020.More news: Resident Evil 3 Remake LEAKS: Nemesis for PS4 appears on PlayStation Store
Brent crude, the global benchmark, rose $1.13 to $61.62 a barrel by 1426 GMT.
"If OPEC and Russian Federation don't extend and deepen their cuts, we could see Brent Blend dip to the $40s next year for a shorter period", Tonhaugen said.
Vitol Group, the world's largest independent oil trader, expects US production to increase by 700,000 barrels a day from December 2019 to December 2020, compared to growth of 1.1 million barrels a day from the end of 2018 to the end of 2019.
Iraq, the second-largest oil producer within Opec, said it supports any goals to rebalance the oil market and lead to its stability. However, if the actual effect of the IMO rules on crude demand turns out to be zero the "call on OPEC" - the amount of OPEC oil needed to meet demand - drops by 1.9 million bpd year-on-year to 28.3 million bpd. "Alas, without deeper cuts taking effect in January 2020, large global implied stock builds are on the cards", Tonhaugen remarked. The additional 1.4 million bpd of "cuts" are owed entirely to involuntary declines from Iran and Venezuela, both of which are exempt from the agreement.More news: World first’ cell phone detection cameras rolled out in Australia
Saudi Arabia's stock market rose in early trade on Tuesday on the back of rising oil prices, while Dubai and Abu Dhabi were shut for public holidays.
Saudi officials, including Prince Abdulaziz, have insisted on stricter compliance with the current cuts, especially as countries such as Iraq and Nigeria have produced well above their quotas while Riyadh has cut more than demanded.
The challenge for OPEC+ is the strong supply growth elsewhere in the world.
However, RBC Capital Markets said there's no widespread enthusiasm for a deeper reduction among OPEC members at this time, though if prices were to move appreciably lower in the immediate run-up to the meeting, a cut in the 1.6 -1.8 million bpd range could come under serious consideration.More news: Trump says Macron has insulted North Atlantic Treaty Organisation alliance