US benchmark West Texas Intermediate for August delivery was up 12 cents at $61.13 a barrel and Brent climbed 11 cents to $64.56 in late-morning trade.
Both contracts also ended higher on Tuesday as traders bet on a fall in stockpiles when the US Department of Energy releases its weekly report later today, indicating strong demand in the world's biggest economy.
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The closely watched Markit Economics Composite Purchasing Managers Output Index (PMI) came in at 54.1 points in June, up from 53.6 in May, for its best performance since May 2011.
Any reading above 50 points suggests expansion.
Eurozone finance ministers will meet today to work on details of a new bailout for Greece after it submitted proposals at the start of the week, fanning hopes for a breakthrough after five months of stalemate.
A full European Union summit is slated for tomorrow.
Greece is up against a deadline of June 30 to repay the International Monetary Fund around 1.5 billion euros ($1.7 billion). Failure to pay will see it default and possibly crash out of the eurozone, or even the EU.
However, Bernard Aw, market strategist at IG Markets Singapore, warned, "While the mood for the negotiations has definitely turned positive and constructive, I think it may be premature to pin too much hopes on a 'done deal'."
He said Greek Prime Minister Alexis Tsipras still needed to convince his anti-austerity party to approve concessions needed to unblock bailout funds.
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