Brent crude futures for September fell 21 cents to USD 66.51 a barrel.
Investors will also monitor trade-related developments, as well movement in the rupee and crude oil prices, they added.
Oil companies in the Gulf of Mexico had cut more than 1 million barrels per day (bpd) of output.
Analysts said the rise in crude oil futures was largely in tandem with a firming trend in global trade.
Crude oil contracts for July delivery traded lower by Rs 35, or 0.88 per cent, at Rs 3,943 per barrel.
Oil prices are being pressured by ongoing worries about demand as the US-China trade war.
On the Multi Commodity Exchange, crude for delivery in July contracts was trading lower by Rs 58 at Rs 3,889.
During April-May, the eight sectors grew by 5.7 per cent compared to 4.4 per cent in the same period last year.
OPEC+ members agreed to curb oil output by 1.2 million barrels per day from January 1.
On MCX, crude oil for delivery in July contracts was trading higher by Rs 64, to Rs 4,100 per barrel.