Concerns about an "uncontrollable" Greek economic crisis and indications of strong demand pushed crude oil prices well into positive territory early Wednesday.

Brent crude oil prices climbed 1.8 percent above the previous close to $64.90 per barrel early in the Wednesday session. West Texas Intermediate, the U.S. benchmark, rallied 1.6 percent to $60.94 per barrel.

The early session rally was sparked in part by indications from the American Petroleum Institute that U.S. crude oil stockpiles dropped for the seventh straight week in a sign of strong demand, despite the sputtering of a U.S. economy slow to move out of the wake of the global financial crisis.

Markets may be reacting in anticipation of actions from Federal Reserve Chair Janet Yellen, who addresses the U.S. economic momentum in a mid-afternoon meeting in Washington.

Yellen in May warned the U.S. economy was still facing headwinds. The U.S. Energy Information Administration publishes its data for the week ending June 12 late Wednesday.

Crude oil prices have been relatively stable for the better part of a month amid signs that some of the oversupply was vanishing from a global economy still struggling to gain traction.

On the supply side, EIA said in its latest drilling productivity report that output from the U.S. shale basins behind the glut is expected to drop off starting in July.

Overseas, the Bank of Greece issued a stark warning of a pending disaster should leaders fail to reach a financial agreement with European and international lenders.

"A manageable debt crisis, as the one that we are currently addressing with the help of our partners, would snowball into an uncontrollable crisis, with great risks for the banking system and financial stability," the bank warned in a statement.