* U.S. rig count rises back to 800 -Baker Hughes
Middle East tensions prevent further price drops
* Falling Venezuelan output seen as risk to supply (Adds Rosneft output figures, comment, updates prices)
By Henning Gloystein
SINGAPORE, March 19 (Reuters) - Oil prices fell on Monday as increased drilling in the United States pointed to more output, raising concerns about a return of oversupply.
U.S. West Texas Intermediate (WTI) crude futures were at $62.14 a barrel at 0739 GMT, down 20 cents, or 0.3 percent, from their previous close.
Brent crude futures were at $65.99 per barrel, down 22 cents, or 0.3 percent.
Monday's price falls in part reversed increases last Friday, which came on concerns over tensions in the Middle East.
On a simple supply versus demand basis, however, oil markets are facing the possibility of a renewed glut after being in a slight deficit for much of last year.
U.S. drillers added four oil rigs in the week to March 16, bringing the total count to 800, the weekly Baker Hughes drilling report said on Friday.
"Surging U.S. production will hamper exponential growth in crude oil prices," Singapore-based brokerage Phillip Futures said on Monday.
The U.S. rig count, an early indicator of future output, is much higher than a year ago as energy companies have boosted spending.
Thanks to the high drilling activity, U.S. crude oil production <C-OUT-T-EIA> has risen by more than a fifth since mid-2016, to 10.38 million barrels per day (bpd), pushing it past top exporter Saudi Arabia.
Only Russia produces more, at around 11 million bpd, although U.S. output is expected to overtake Russia's later this year as well.
Soaring U.S. output, as well as rising output in Canada and Brazil, is undermining efforts led by the Organization of the Petroleum Exporting Countries (OPEC) and Russia to curb supplies and bolster prices.
Amid Russia's efforts to restrain output, Russian oil giant Rosneft said on Monday that its fourth quarter 2017 liquid hydrocarbon production reached 56.51 million tonnes, raising its full-year output by 7.3 percent to 225.5 million tonnes, or 4.53 million bpd.
Many analysts expect global oil markets to flip from slight undersupply in 2017 and early this year into oversupply later in 2018.
"Let's face it, there is still too much oil," said Matt Stanley, a fuel broker with Freight Investor Services in Dubai in a note.
One risk seen to supplies, however, is Venezuela.
The International Energy Agency said last week that Venezuela, where an economic crisis has cut oil production by almost half since early 2005 to well below 2 million bpd <PRODN-VE>, was "clearly vulnerable to an accelerated decline," and that such a disruption could tip global markets into deficit despite soaring U.S. output.
(Reporting by Henning Gloystein; editing by Joseph Radford and Richard Pullin)