Analysts said the rise in crude oil futures was largely in tandem with a firming trend in global trade.
Brent crude futures were up USD 1.17, or 1.9 per cent, at USD 63.10 a barrel by 0505 GMT.
The IEA is reducing its 2019 oil demand forecast due to a slowing global economy amid a US-China trade spat.
On the Multi Commodity Exchange, crude oil for delivery in July contracts shed Rs 8, or 0.2 per cent, to Rs 3,908 per barrel.
Analysts said the rise in crude oil futures was largely in tandem with a firm trend in global market.
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.