Brent crude futures, the world record, ended the trading session up $ 2.95, or 7.48%, to settle at $ 42.40 a barrel.
US West Texas Intermediate crude futures jumped $ 3.15, or 8.48%, to settle at $ 40.29 a barrel.
The two benchmarks jumped more than four dollars earlier in the day.
Oil markets rose after Saudi Arabia indicated that it and other crude producers may amend their current agreement to reduce supplies and may withhold more barrels from the market if demand falls in winter with the increase in virus infections and before the vaccine is widely available.
On the other hand, OPEC members fear that US President-elect Joe Biden will ease sanctions on Iran and Venezuela, which may mean an increase in oil production and will make it difficult to find a balance between supply and demand.