Brent crude was up 87 cents, or 3.8pc, at $23.63 a barrel by 1106 GMT after closing on Monday at $22.76, its lowest finish since November 2002.
US crude was up $1.45, or 7.2pc, at $21.54 a barrel after settling in the previous session at $20.09, its lowest since February 2002.
Oil markets have faced a double whammy from the coronavirus outbreak and a race to win market share between Saudi Arabia and Russia after OPEC and other producers failed this month to agree on deeper supply cuts to support oil prices.
Trump and Putin agreed in a phone call to have their top energy officials discuss stabilising oil markets, the Kremlin said on Monday.
Although the futures market is seeing a recovery, physical cargoes are selling in some regions at single digits, with sellers offering hefty discounts.
"The gap between physical assessments and futures reflects the differences between the realities on the ground and speculation about efforts to ease that pressure going forward," JBC Energy said.
With a plunge in prices that has knocked about 60pc off oil prices this year, a commissioner with the Texas state energy regulator renewed a call for restrictions on crude production because of a national supply glut.
In a sign of how well the market is supplied, the front-month Brent futures contract for May is trading at a discount of $13.95 a barrel to the November contract, the widest contango spread ever seen.
A contango market implies traders expect oil to be higher in the future, encouraging them to store oil now to sell later.
Saudi Arabia, de facto leader of the Organization of the Petroleum Exporting Countries (OPEC), plans to boost its oil exports to 10.6 million barrels per day (bpd) from May on lower domestic consumption, a Saudi Energy Ministry official said.
Global oil refiners, meanwhile, have cut their throughput because of the slump in demand for transportation fuel, with European refineries reducing output by at least 1.3 million bpd, sources told Reuters.
Exxon Mobil Corp closed a small crude distillation unit at its 502,500 bpd Baton Rouge refinery in Louisiana because of low demand, sources said.
The chief economist for global commodities trader Trafigura said that oil demand could fall in the coming weeks by as much as 30pc from consumption at the end of last year.
A Reuters survey of 40 analysts forecast Brent crude prices would average $38.76 a barrel in 2020, 36pc lower than the $60.63 forecast in a February survey.