* U.S. crude stocks drop by nearly 11 mln bbls -EIA
* Britain gains initial support for EU-led mission in Hormuz
* Sluggish demand outlook weighs on prices (New throughout, updates prices, market activity, comments)
By Stephanie Kelly
NEW YORK, July 24 (Reuters) - Oil prices steadied on Wednesday, failing to draw much support from a large decrease in U.S. crude stockpiles as investors also worried about global oil demand.
Brent crude futures rose 23 cents to $64.06 a barrel by 12:04 p.m. EDT (1604 GMT), while U.S. West Texas Intermediate (WTI) crude futures fell 7 cents to $56.70 a barrel.
Earlier in the session, the front-month Brent contract flipped to trade at a discount to the second-month contract, a market structure known as contango, for the first time since March. Sentiment in the oil market has darkened as investors worry about slowing global economic growth weakening demand for oil.
Prices initially gained after Energy Information Administration data on Wednesday showed a large drawdown in U.S. crude stockpiles. Crude inventories fell by 10.8 million barrels in the week to July 19. Analysts expected a decrease of 4 million barrels.
But prices later pared gains.
"The market is going to try to say that (the drawdown) was probably due to (Hurricane Barry), and so the market is not overreacting to it," said Phil Flynn, an analyst with Price Futures Group in Chicago.
U.S. oil companies cut some production in the Gulf of Mexico ahead of Hurricane Barry, which came ashore in Louisiana earlier this month. Prices received some support from geopolitical risk premium on heightened tensions in the Middle East.
A U.S. Navy ship took defensive action against a second Iranian drone in the Strait of Hormuz last week, but did not see the drone go into the water, the U.S. military said on Tuesday.
Iran's president, Hassan Rouhani, said on Wednesday his country was ready for "just" negotiations but not if they meant surrender.
Also fueling tensions, Britain gained initial support from France, Italy and Denmark for its plan for a European-led naval mission to ensure safe shipping through the Strait of Hormuz following Iran's capture of a British-flagged tanker.
The military adviser to Iran's supreme leader was quoted on Wednesday as saying that any change in the status of the Strait of Hormuz, which Tehran says it protects, would open the door to a dangerous confrontation.
Reporting by Stephanie Kelly in New York Additional reporting by Ahmad Ghaddar in London, Roslan Khasawneh in Singapore and Aaron Sheldrick in Tokyo Editing by Jan Harvey, Matthew Lewis and David Gregorio