Australia's S&P/ASX, Japan's Nikkei and Hong Kong's Hang Seng all decline.
WTI crude for delivery in August fell 1.19% at $58.76 per barrel, while Brent crude was trading down 0.83% at $63.06.
The number of US rigs actively drilling for oil has dropped for 27 successive weeks.
Gold prices to take their cues from developments in Greece and from Fed speak on US interest rates.
Analysts warn that worst may not be over, given the robust stock markets and the likelihood of higher US interest rates.
In Norway, petrol prices are 26% higher than that in the UK, where price per litre is £1.21.
Break of current resistance will open doors to new multi-month highs.
Shale oil boom in US is expected to come to halt amid oil price cartel Opec's unchanged production target.
WTI crude fell 0.83% to $58.64 per barrel, while Brent crude for July fell 0.68% to $62.88 as at 1.10am ET.
Capital Economics believes that Brent is "now back within a whisker" of its end-2015 forecast of $60 per barrel.
Oil hit its lowest price in six years in January, $45 a barrel, but has since climbed up to trading near $62.
With its ever-increasing oil demand, India is expected to become prime customer for Opec members.
Hasan Qabazard says Brent crude will trade between $40 to $50 a barrel in the fourth-quarter of 2015.
Commerzbank expects that Opec will confirm its production target of 30 million barrels per day on 5 June.
Petrobras is the operator and has a 75% stake in the prospect, with India's ONGC holding 25%.