Oil prices fell below $63 a barrel today, weighed down by concerns of a growing supply glut, paring earlier gains spurred by hopes that Greece might avert a default.

Oil prices retreated shortly after the opening of trading in New York, highlighting the pressure on the market from a crude oil supply overhang, particularly in the Atlantic basin.

Brent crude for August delivery was down 32 cents at $62.70 a barrel by 9.20 a.m. EDT, after hitting a session high of $63.74 a barrel.

Front month U.S. crude was 54 cents lower at $59.07 a barrel, more than a dollar below highs of $60.30 a barrel.

Hopes for a last-minute Greek debt deal boosted global financial markets after Greece's latest proposals to resolve its debt crisis got a positive reaction in Brussels.

"The Physical market is very weak," Oliver Jakob, analyst at Zug, Switzerland-based Petromatrix, said. "If the physical markets are any indication, you would expect to see further declines."

Around 10 million barrels of unsold crude, mainly from Nigeria, are held in offshore storage despite strong summer demand, Morgan Stanley said in a research note on Monday, potentially creating a negative outlook for oil in the second half of the year.

"If there are this many challenged cargoes in this strong demand environment, we worry about the outlook for physical oil this fall (autumn) when crude runs and gasoline demand fall seasonally," the note said.

High domestic U.S. oil production, which has held up at around 9.6 million barrels a day, the highest level since the early 1970s, continued to weigh on prices.

U.S. oil producers added a rig each in the key Permian and Bakken shale basins last week, fuelling worries over high domestic oil output, even as the total number of active U.S. rigs fell last week, data on Friday showed.

Although U.S. oil production was expected to decline slightly between the second and third quarters of 2015, output

"would continue to grow in 2016 by 150,000 barrels per day at the current rig count," Goldman Sachs said in a note.