Oil prices started the last full trading week in June in stable territory as the Greek debt crisis continues, but prices are still below June 1 levels.

The price for Brent crude oil was down a fraction of a percent in early Monday trading to $62.88 per barrel. West Texas Intermediate, the U.S. benchmark, lost modest ground as well, trading at $59.45 per barrel. Brent is off 3.4 percent from June 1, though WTI is faring better, down 1 percent from the start of the month.

Crude oil prices have been relatively stable amid signs that global economic recovery was erasing some of the oversupply scenarios behind the early 2015 market decline.

Last week, the American Petroleum Institute said U.S. petroleum deliveries, a measure of demand, in May grew 5.3 percent year-on-year to 19.5 million barrels per day for its highest level of the year. With low prices at the pump, and good job numbers in the United States, API said much of the demand was driven by gasoline.

A drilling report from the U.S. Energy Information Administration said U.S. shale oil production could start declining by June. At this month's conference, members of the Organization of Petroleum Exporting Countries, meanwhile, said their output would remain steady amid an expected increase in global demand for oil.

European and international lenders are trying to persuade Greek leaders to enact a series of economic reforms in order to stave off a spiraling debt crisis. Louka Katseli, the head of the National Bank of Greece, told the BBC during the weekend banks were weathering the storm, though the situation could turn severe without a comprehensive agreement.