Since spring when the Trump Administration said it would impose sanctions on Iran, crude traders have priced in a risk premium reflecting the supply shortages that may occur when exports from the third-largest OPEC member are cut. As the Nov. 4 date for imposing sanctions draws nearer, the premium has increased.
"The fear is that the sanctions could be so successful that it takes more oil off the market than the OPEC and non-OPEC producers can make up for," said Andrew Lipow, president of Lipow Oil Associates in Houston.
Brent crude futures rose $1.67 to $79.04 a barrel, a 2.2 percent gain, by 1:10 p.m. EDT (1710 GMT).
US West Texas Intermediate (WTI) crude futures gained $1.95, or 2.9 percent, to $69.49 a barrel.
Washington has told its allies to reduce imports of Iranian oil and several Asian buyers, including South Korea, Japan and India appear to be falling in line.
But the US government does not want to push up oil prices, which could depress economic activity or even trigger a slowdown in global growth.
US Energy Secretary Rick Perry met Saudi Energy Minister Khalid al-Falih on Monday in Washington, as the Trump administration encourages big oil-producing countries to keep output high. Perry will meet with Russian Energy Minister Alexander Novak on Thursday in Moscow.
Russia, the United States and Saudi Arabia are the world's three biggest oil producers by far, meeting around a third of the world's almost 100 million barrels per day (bpd) of daily crude consumption.
Russian Energy Minister Alexander Novak said on Tuesday that Russia and a group of producers around the Middle East which dominate the Organization of the Petroleum Exporting Countries may sign a new long-term cooperation deal at the beginning of December, the TASS news agency reported. Novak did not provide details.
A group of OPEC and non-OPEC producers have been voluntarily withholding supplies since January 2017 to tighten markets, but with crude prices up by more than 40 percent since then and markets significantly tighter, there has been pressure on producers to raise output.
On Tuesday the US Energy Information Administration cut its 2018 world oil demand growth forecast by 80,000 barrels per day to 1.58 million bpd.
US crude inventories were forecast to have fallen for a fourth consecutive week last week, according analysts polled ahead of reports from industry group the American Petroleum Institute (API) at 4:30 p.m. EDT (2030 GMT) and the US Department of Energy on Wednesday.
Also supporting prices was an attack on the headquarters of Libya's National Oil Corporation (NOC) in the capital Tripoli on Monday.
The NOC has continued to function relatively normally amid chaos in Libya. Oil production has been hit by attacks on oil facilities and blockades, though last year it partially recovered to around one million barrels per day.
As Middle East markets tighten, Asian buyers are seeking alternative supplies, with South Korean and Japanese imports of US crude hitting a record in September.
US oil producers are seeking new buyers for crude they used to sell to China before orders slowed because of the trade disputes between Washington and Beijing.