* Firmer dollar caps market; equities support sentiment
* Euro falls ahead of EU summit on Greek debt crisis
* Coming Up: U.S. weekly oil inventory data (2030 GMT)
(Updates detail, prices, comment)
By Christopher Johnson
LONDON, March 23 (Reuters) - Oil fell towards $81 a barrel on Tuesday, under pressure as the dollar strengthened ahead of a European Union summit to discuss Greece's debt crisis.
Forecasts of a build in U.S. crude oil stocks in a weekly inventory report due later in the day also undermined sentiment despite a rise in stock markets in Europe and the United States.
U.S. crude for May delivery <CLc1> was down 33 cents at $81.27 a barrel by 1217 GMT. The April delivery contract expired on Monday, settling up 57 cents at $81.25.
London Brent crude for May <LCOc1> fell 25 cents to $80.29.
"The market is expecting a build in crude inventories and the dollar is also a negative factor," said Christopher Bellew, broker at Bache Commodities in London.
Oil prices have traded above $83 on four occasions this month, but have failed each time to hold the gains, partly due to expectations of an oversupply of oil this year.
World oil demand is still well below its level in 2007, before record prices and recession slashed consumption and global oil inventories are well above their five-year average.
The strength of the dollar has helped depress oil as concern over Greece's debt crisis has pushed down the euro. Commodities often move inversely to the dollar as they are priced in the U.S. currency on international markets.
The euro fell against the dollar on Tuesday as uncertainty over possible support for Greece continued in the run-up to an EU summit later this week. [
]
HELP FOR GREECE
Concerns that the EU may not reach a decision this week kept the euro close to a three-week low hit on Monday and fanned risk aversion, supporting safe-haven flows into the U.S. dollar.
The dollar index <.DXY>, a measure of its performance against six other currencies, was up around 0.4 percent at 80.961 at 1200 GMT, but still well below an eight-month high of 81.342 set in February.
EU leaders will discuss the issue of a financial aid package for Greece at a summit in Brussels on March 25-26, after Athens said it could not deliver promised deficit cuts if its borrowing costs remained so high, and that it may have to seek help from the International Monetary Fund.
The American Petroleum Institute (API) was due to unveil its weekly inventory report at 2030 GMT on Tuesday and was expected to say U.S. crude oil stocks rose 1.3 million barrels last week, a preliminary Reuters poll showed.
Distillate stocks were expected to have eased by 1.1 million barrels, while gasoline stocks were projected to fall 1.6 million barrels. [
]Oil shrugged off a gain on European stock markets.
European shares rose in early trade on Tuesday, with miners climbing as gold <XAU=> prices picked up. By 1200 GMT, the FTSEurofirst 300 <
> index of top European shares was up 0.34 percent at 1,068.78 points, after three days of losses."Equity markets are comfortably higher this morning and they are supporting sentiment in all commodities," said Eugen Weinberg, analyst at Commerzbank in Frankfurt.
"But the dollar is stronger, which is a negative factor, as are the fundamentals of supply and demand, which are not good."
The market awaited data at 1400 GMT from the U.S. National Association of Realtors, which was due to release existing home sales for February. Economists forecast a total of 5.00 million annualised units versus 5.05 million in January. (Editing by Keiron Henderson)