Brent crude futures were trading down 40 cents, or 0.7 per cent, at 60.54 dollars a barrel by 0701 GMT.
Tensions in the Middle East after last week's tanker attacks remain high, with Trump saying he was prepared to take military action to stop Iran having a nuclear bomb but leaving open whether he would sanction the use of force to protect Gulf oil supplies.
A USA official this morning claimed an American drone was taken down in worldwide airspace over the Strait of Hormuz by an Iranian surface-to-air missile. Extending the cuts would shrink global inventories by about 500,000 barrels a day in the second half of this year, according to an OPEC committee, helping boost prices. The buying started earlier in the week on the hopes that a meeting between U.S. President Trump and China President Xi Jinping will lead to an eventual trade deal between the two economic powerhouses.
Followed by Trump's tweet, crude prices had begun to gather pace despite a stubborn rise in United States inventories, while sending more 1000 USA troops to Middle East had raised red flags over the Gulf and added to concerns of potential supply disruption through the strategically significant Strait of Hormuz.
U.S. West Texas Intermediate (WTI) crude futures were up 79 cents, or 1.5%, at $54.55 a barrel.
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The EIA report also showed that gasoline inventories were down 1.7 million barrels, while distillate stockpiles edged lower by 600,000 barrels last week.
Refined products also posted surprise drawdowns due to a rise in refining and crude exports, as well as a drop in crude production.
That and the bullish EIA report "all begin to reduce the uncertainties for an oil market that has a whole bunch of them, maybe the biggest of which is where demand is going to go", said Rob Thummel, managing director at Tortoise, a Kansas-based money manager that oversees $21 billion in assets.
Fears of a confrontation between Iran and the United States have mounted, with Washington blaming Tehran for the tanker attacks and Tehran denying any role.
Equities held steady after the U.S. Federal Reserve's decision to hold interest rates steady, as expected, after concluding a two-day policy meeting on Wednesday. "I don't think it's more than a sentiment thing along those lines". -China trade deal and on the potential for conflict between the USA and Iran in the Middle East after tanker attacks there last week.