Brent crude futures were up 23 cents, or 0.4%, at $63.23 a barrel by 0054 GMT. Prices rose 1.6% last week. US West Texas Intermediate crude was down by 12 cents, or 0.2%, at $56.08 a barrel. WTI gained 1% last week.
Brent crude prices have risen over 13.5% from $66.11 per barrel on March 15 to $75.25 per barrel on April 25, while retail petrol prices in Delhi have gone up by just 0.5% from Rs 72.55 per litre to Rs 72.95 per litre.
Since the beginning of FY19, crude has been caught up in wider financial market weakness as higher US interest rates, the US-China trade dispute and the US shutdown unnerved investors and exacerbated worries over global growth.
Front-month Brent crude oil futures were at $72.63 per barrel, 0.25 percent from their last close. Signs of slowing economic growth and lower fuel demand increases, especially in Asia's large emerging markets are weighing on the oil markets.