By early afternoon in Europe, benchmark oil for November delivery was up $2.25 to $82.49 a barrel in electronic trading on the New York Mercantile Exchange. The contract rose 39 cents to finish at $80.24 per barrel in New York on Monday.
In London, Brent crude for November delivery rose $1.45 at $105.39 on the ICE Futures exchange.
Energy prices followed gains in stock markets on confidence that European finance ministers might soon take action to stem the region's sovereign debt crisis.
For weeks, investors have been worried that Europe's problems could lead to another recession, which would cut demand for oil and gasoline. Demand for gasoline in the U.S. is already below year-ago levels.
"The main reasons are a change of sentiment on financial markets, reflected in rising equity markets and a somewhat weaker U.S. dollar," said a report on commodities from Commerzbank in Frankfurt. "The downside risks for the economy and oil demand have not disappeared overnight, so a renewed price fall cannot be ruled out."
Analysts were also looking to consumer confidence data expected later Tuesday out of the U.S. for clues on the future strength of demand for fuel. Oil prices have slumped since spring as concern has grown about the global economy, but signs of improving consumer sentiment could boost crude.
"As for today all eyes will likely be on September's consumer confidence index," which is expected to rise to 46.0 from 44.5 in August, energy trader and consultant The Schork Group said in a report.
Investors are also awaiting new information on U.S. stockpiles of crude and refined products.
Data for the week ending Sept. 23 is expected to show no change in crude oil stocks, while gasoline stocks are seen rising by 1.2 million barrels, according to a survey of analysts by Platts, the energy information arm of McGraw-Hill Cos.
The American Petroleum Institute will release its report on oil stocks later Tuesday, while the report from the Energy Department's Energy Information Administration -- the market benchmark -- will be out on Wednesday.
In other energy trading, heating oil rose 4.35 cents to $2.8465 per gallon, gasoline futures advanced 5.23 cents to $2.5807 per gallon and natural gas rose 5.8 cents to $3.84 per 1,000 cubic feet.
Pamela Sampson in Bangkok contributed to this report.