UPDATE 4-Oil prices rise after big draw in U.S. stockpiles

* U.S. crude stockpile draw supports prices

* Trump could pull out of global climate accord (Updates throughout, changes dateline, previous TOKYO)

LONDON, June 1 (Reuters) - Oil prices rose on Thursday, rallying from three-week lows after industry data showed a big draw in U.S. crude stocks, suggesting that the world's largest oil market could be tightening faster than expected.

Brent crude oil was up 40 cents at $51.16 a barrel by 0940 GMT, while U.S. light crude gained 40 cents to $48.72.

Both crude benchmarks fell by about 3 percent to three-week lows on Wednesday after news that an increase in Libyan oil production helped to boost monthly OPEC crude output in May, the first monthly rise this year.

But industry data on U.S. oil inventories from the American Petroleum Institute (API) late on Wednesday helped the market to pare those losses.

API figures showed that U.S. crude inventories fell by 8.7 million barrels to 513.2 million in the week to May 26, compared with analyst expectations for a decrease of only 2.5 million barrels.

"This was well ahead of forecasts," said Stephen Brennock, analyst at London brokerage PVM Oil Associates. "(It) is helping the oil market regain some ground this morning."

The U.S. Energy Information Administration (EIA) was due to report its official figures for U.S. stockpiles at 1500 GMT on Thursday and investors were waiting to see if the API figures were confirmed.

The U.S. inventories data provided some relief after a week of negative news on the global supply-demand balance.

The Organization of the Petroleum Exporting Countries and other producers including Russia are trying to restrict output to drain stockpiles that are close to record highs in many parts of the world.

But U.S. crude production is rising fast as new technology helps to extract shale oil, making the United States more self-sufficient in energy.

President Donald Trump has vowed to provide extra support for U.S. oil production and is widely expected to pull the United States out of a landmark global climate accord.

Trump has said he will announce later on Thursday a decision on whether to keep the United States in the global pact to fight climate change. A source close to the matter has said he is preparing to withdraw from the Paris agreement.

Phillip Futures' investment analyst Jonathan Chanes said that a U.S. withdrawal would signal Trump's intention to further roll back emission regulations.

"That would favor the use and demand of fossil fuels, thus giving a much-needed boost to oil prices,"" Chanes said.

(Additional teporting by Aaron Sheldrick in Tokyo; Editing by David Goodman)