Brent has rallied more than 3 percent after touching a four-month low last week, while U.S. crude is up nearly 5 percent from last Thursday's eight-week low.
U.S. West Texas Intermediate crude for delivery in September jumped 92 cents, or 1.4 percent, to end the session at $67.35. The contract expires on Tuesday.
Brent crude futures rose 51 cents to $72.72 a barrel by 2:24 p.m. ET. The October WTI contract, the most heavily traded, was up 56 cents, or nearly 1 percent, at $65.98 a barrel.
September WTI's premium to the October contract widened to as much as $1.69 a barrel on Tuesday, after narrowing since the beginning of August. The front-month spread widened significantly in July after an unexpected outage at Canada's oil sands facility dented flows of crude into Cushing, Oklahoma, the delivery point for U.S. futures.
Total U.S. crude stockpiles, however, were forecast to have drawn down about 1.5 million barrels last week, according to analysts polled ahead of industry data due at 4:30 p.m. EDT (2030 GMT) on Tuesday and government data on Wednesday.
Oil prices have gained in the last two sessions following weeks of declines on the prospect of lower oil supply from Iran. The United States is trying to halt Iranian oil exports in an effort to force Tehran to negotiate a new nuclear agreement and to curb its influence in the Middle East.
However, the full impact of the Iran sanctions are unclear.
While most of Europe's energy firms are likely to fall in line with Washington, China has indicated that it will continue to buy Iranian oil.
BNP Paribas said it expected oil production from the Organization of the Petroleum Exporting Countries, of which Iran is a member, to fall from an average of 32.1 million barrels per day (bpd) in 2018 to 31.7 million bpd in 2019.
However, oil exports from southern Iraq are on course to hit another record high this month, two industry sources said, adding to signs that OPEC's second-largest producer is following through on the group's agreement to raise output.
The market also continued to eye the U.S.-China dispute that threatens to undermine global growth and, therefore, consumption of industrial commodities.
A Chinese delegation is due in Washington this week to try to resolve the dispute, but U.S. President Donald Trump told Reuters on Monday he does not expect much progress and that resolving the disagreement will "take time".
"The upcoming U.S.-China trade talks are unlikely to offer any significant breakthroughs as more formal discussions and, hence, decisions will likely await expected talks in November between Trump and Xi," Jim Ritterbusch, president of Ritterbusch and Associates, said in a note.
— CNBC's Tom DiChristopher contributed to this report.