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* OPEC, allies extend output curbs until March 2020
* U.S. crude oil inventories fall 5 mln bbls -API
* Morgan Stanley lowers long-term Brent price forecast
* Upcoming: U.S. EIA data at 1430 GMT (Updates prices, adds context)
LONDON, July 3 (Reuters) - Oil prices rebounded on Wednesday after a steep fall in the previous session when OPEC and its allies' decision to extend output cuts was not enough to counter investors' concerns about the slowing global economy.
Prices were supported by a strong U.S. stock market opening and widely watched data showing a larger-than-expected drawdown in U.S. crude oil inventories, with government data due later in the day.
Brent crude futures for September delivery were trading up 95 cents at $63.35 a barrel by 1350 GMT.
U.S. crude futures for August were up 76 cents at $57.01 a barrel. Both benchmarks fell more than 4% on Tuesday on worries about a global economic slowdown.
The Organization of the Petroleum Exporting Countries and other producers such as Russia, a group known as OPEC+, agreed on Tuesday to extend oil supply cuts until March 2020 as members overcame differences to try to prop up prices.
"We had a pretty sharp correction yesterday so after that, a little rebound is expected. Globally, the market is concerned about oil demand growth potential," Olivier Jakob of Petromatrix consultancy said.
"Extending the cut by six or nine months, it doesn't really matter if the level stays the same. If you (OPEC) really wanted to target stock levels, you would need deeper cuts but Saudi Arabia has already gone beyond its cut target."
Ahead of government data due later on Wednesday, industry group the American Petroleum Institute (API) said that U.S. crude inventories fell by 5 million barrels last week, more than the expected decrease of 3 million barrels.
The OPEC+ agreement to extend oil output cuts for nine months should draw down oil inventories in the second half of this year, boosting oil prices, analysts from Citi Research said in a note.
"Keeping cuts through the end of 1Q aims to avoid putting oil into the market during a seasonal low for demand and refinery runs," they said.
Still, signs of a global economic slowdown hitting oil demand growth worried investors after global manufacturing indicators disappointed and the United States opened another trade front after threatening the EU with more tariffs.
The S&P 500 index hit a record high at the open on Wednesday, boosted by healthcare stocks, as bets of an interest rate cut were spurred by the fears of a slowing global economy due to simmering trade tensions.
The U.S. trade deficit jumped to a five-month high in May and the ADP National Employment Report on Wednesday showed that an increase in private payrolls was well below economists' expectations.
Barclays expects demand to grow at its slowest pace since 2011, gaining less than 1 million barrels per day year-on-year this year.
Morgan Stanley, meanwhile, lowered its long-term Brent price forecast on Tuesday to $60 per barrel from $65 per barrel, and said the oil market is broadly balanced in 2019.
Crude prices were also capped by signs of a recovery in oil exports from Venezuela in June and growth in oil production in Argentina in May. (Additional reporting by Jessica Jaganathan in Singapore; Editing by Joseph Radford/Louise Heavens/Susan Fenton)