By Amrith Ramkumar

Oil prices rose Tuesday, continuing a recent rebound with analysts anticipating that crude producers led by Saudi Arabia and Russia will extend supply curbs in response to the energy industry's crisis.

U.S. crude futures for July delivery surged 3.9% to $36.81 a barrel on the New York Mercantile Exchange. Front-month futures have risen to their highest level since early March after dropping below $0 for the first time ever on April 20 due to a lack of available storage for excess crude. Drivers returning to the road as states and countries lift coronavirus lockdowns and record production cuts are fueling the rebound.

Tuesday's moves came after the Organization of the Petroleum Exporting Countries and allies including Russia moved a conference call to discuss future output cuts to Thursday from June 9. The group is close to extending the supply cuts through Sept. 1, a move that some analysts say would lend further support to oil prices as demand continues to climb.

Still, prices tend to be very volatile around OPEC discussions, and some analysts are skeptical that Russia and some cartel members will fully comply with any agreement. Saudi Arabia and Russia had a production feud earlier in the year that raised global output as demand crashed. A landmark April agreement to cut supplies ended that spat.

Investors are also keeping a close eye on production in North America, where many companies have shut in productive wells in response to ultralow prices. If prices continue climbing, those companies would likely try to increase supply, another factor that analysts say could slow oil's momentum.

"A continuation of the oil-price rally in [the second quarter] may prove self-defeating," Giovanni Staunovo, commodity analyst at UBS Wealth Management, said in a note.

Brent crude for August delivery, the global gauge of oil prices, advanced 3.3% to $39.57 a barrel on the Intercontinental Exchange on Tuesday.

The speed of oil's rebound has surprised many market watchers. Demand has returned quickly in China, the world's largest consumer of raw materials, and traffic is gradually climbing in parts of the U.S. and Europe.

Many analysts expect the recovery will be bumpy. Oil prices have risen faster than gasoline prices, giving refiners that turn crude into fuel products less incentive to take in more oil. At the same time, a flood of oil from Saudi Arabia has been making its way to large consumers such as the U.S. and China.

A rise in imports from the kingdom led to a big increase in U.S. crude-oil stockpiles for the week ended May 22. That rise reversed a recent drop in inventories that had also buoyed prices by signaling that a global glut was easing. Figures for last week are scheduled to be released Wednesday.

Analysts were also monitoring the path of Tropical Storm Cristobal, which has formed in the southern Gulf of Mexico. The storm is expected to move north in the coming days, and any possible disruptions to energy industry activities in the region could further lower crude supplies.

Write to Amrith Ramkumar at amrith.ramkumar@wsj.com