* U.S. crude output highest since 1970s, but crude stocks fall
* U.S. output close to that of top producers Russia, Saudi Arabia
* Saudi Arabia says it will take more time to re-balance markets (Adds comment, updates prices)
SINGAPORE, Dec 21 (Reuters) - Oil prices were stable on Thursday, supported by falling crude inventories in the United States but capped by output that is fast approaching 10 million barrels per day, a level only surpassed by Saudi Arabia and Russia.
U.S. West Texas Intermediate (WTI) crude futures were at $58.16 a barrel at 0756 GMT, up 7 cents from their last settlement.
Brent crude futures, the international benchmark for oil prices, were up 6 cents at $64.54 a barrel.
Both crudes gained around 1 percent during their previous sessions, lifted by official data showing a 6.5 million-barrel fall in U.S. crude inventories <C-STK-T-EIA> in the week to Dec. 15 to 436 million barrels, the lowest level since October, 2015.
Countering this on Thursday was another increase in American crude oil production, while a rise in gasoline stocks pointed to a slowdown in demand.
"The rally which has defined H2 looks to have run out of steam as the year draws to a close, and we might be hard pressed to say where further upside can come from at this point, barring some unforeseen supply outage," JBC Energy said in a note.
The energy minister of Saudi Arabia, the world's top crude exporter and OPEC's de-facto leader, said it would take more time to rein in a global supply overhang, which was created by strong global production increases in the years up to 2015.
"We expect the first few months of 2018 to be either flat or a build (in inventories), as it is typically the case with the seasonality in the oil market," Khalid al-Falih told Reuters on Wednesday.
U.S. oil output is close to breaking through 10 million bpd, undermining efforts led by the Organization of the Petroleum Exporting Countries (OPEC) and Russia to tighten the market through withholding supply this year and next.
U.S. crude production <C-OUT-T-EIA> hit 9.79 million bpd last week, its highest since the early 1970s, the only time American output breached 10 million bpd.
This brings U.S. output close to that of top producers Saudi Arabia and Russia, which pump around 10 million and 11 million bpd, respectively.
JBC Energy said "U.S. crude exports are starting to increase."
Oil traders this week also eyed with interest the passing of a U.S. tax bill, which is seen weighing on crude prices in the longer term.
"The passage of the U.S. tax bill is ... a bearish long-term development for oil and gas markets," Barclays bank said.
"The policies ... are likely to reduce demand for gas and oil and raise supplies ... (as) the tax bill preserves renewable energy tax credits, a tax credit for EVs (electric vehicles), and opens up drilling in the Arctic National Wildlife Refuge."
(Reporting by Henning Gloystein; Editing by Joseph Radford and Tom Hogue)