On Monday, oil prices rose above $63 a barrel after European Union (EU) welcomed new proposals from Greece.

Commerzbank analysts said, "Hopes that agreement could be reached in the debt dispute with Greece at today's EU summit are lending buoyancy to oil prices as the new week begins".

However, high levels of global oil production and stocks are still exerting pressure on crude prices. For August delivery, Brent crude was up 30 cents at $63.32 a barrel by 0822 GMT. Front month US crude was 44 cents higher at $60.05 a barrel.

In a research note on Monday, Morgan Stanley said that despite strong summer demand, around 10 million barrels of unsold crude are held in offshore storage. It is creating negative outlook for oil in the second half of the year.

The note said that they worry about the outlook for physical oil this fall (autumn) when crude runs and gasoline demand fall seasonally.

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

Last week, US oil producers added a rig each in the key Permian and Bakken shale basins. This increased worries over high domestic oil output.

Goldman Sachs said in a note that although US 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.