WTI futures, slumped last week by nearly 2 million barrels to their lowest since December.
Tensions spiked between Iran and the West after Iranian commandos seized a British-flagged oil tanker in the Gulf.
Brent crude futures for September fell 21 cents to USD 66.51 a barrel.
Oil companies in the Gulf of Mexico had cut more than 1 million barrels per day (bpd) of output.
Oil prices are being pressured by ongoing worries about demand as the US-China trade war.
OPEC+ members agreed to curb oil output by 1.2 million barrels per day from January 1.
Oil prices have been squeezing higher on escalating tensions in the Middle East.
India bought about 184,000 barrels per day (bpd) oil from the United States over November 2018 to May 2019.
West Texas Intermediate crude was up 70 cents, or 1.2 per cent, at USD 58.13 a barrel.
Global oil prices rose sharply on fears of a US military attack on Iran that would disrupt flows from the Middle East.