* Global markets drop as key Trump adviser departs
* Gary Cohn opposed Trump plans for steel, aluminum tariffs
* Soaring U.S. crude ouutput, rising stocks also weigh
* U.S. crude output to rise to 11.17 mln bpd by Q4 - EIA
SINGAPORE, March 7 (Reuters) - Oil prices fell on Wednesday, pulled down by weaker sharemarkets after a key advocate for free trade in the U.S. government resigned, triggering concerns that Washington would go ahead with import tariffs and risk a trade war.
Soaring U.S. crude oil production and rising inventories were also weighing on crude prices, traders said.
Gary Cohn, economic adviser to U.S. President Donald Trump, seen as a bulwark against protectionist forces within the government, said on Tuesday he was resigning, triggering a more than 1 percent fall in S&P 500 futures in early Wednesday trade.
Crude oil futures followed suit.
Brent crude futures were at $65.37 per barrel at 0149 GMT, down 42 cents, or 0.6 percent from their previous close.
U.S. West Texas Intermediate (WTI) crude futures were at $62.16 a barrel, down 44 cents, or 0.7 percent.
"The overhang from the Cohn resignation ... could see oil prices move lower during today's session," said Stephen Innes, head of trading for Asia/Pacific at futures brokerage OANDA in Singapore.
A voice for Wall Street in the White House, Cohn's move to resign came after he lost a fight over Trump's plans for hefty steel and aluminum import tariffs.
Major powers, including the European Union and China, have warned that such tariffs could lead to retaliatory action and trigger a global trade war, which could grind to a halt economic growth and, by extension, oil consumption.
Traders said oil prices were also weighed by a reported rise in U.S. crude oil inventories.
Crude inventories rose by 5.661 million barrels in the week to 426.880 million barrels, data from the American Petroleum Institute showed on Tuesday.
Official data by the U.S. Energy Information Administration (EIA) is due to be published later on Wednesday.
Overall, oil supplies are ample despite efforts led by the Organization of the Petroleum Exporting Countries (OPEC) and Russia to withhold output in order to prop up prices.
The EIA on Tuesday made its latest in a series of upward revisions for U.S. crude oil production <C-OUT-T-EIA>, which it now expects to rise by more than 120,000 barrels per day (bpd) to 11.17 million bpd by the fourth quarter of 2018.
That would take the United States past Russia to become the world's biggest oil producer. The U.S. already passed top exporter Saudi Arabia late last year.
For 2019, the EIA forecast a crude production increase of 570,000 bpd to 11.27 million bpd.
(Reporting by Henning Gloystein)