* 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 Venezuela 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 $62.03 a barrel at 0248 GMT, down 31 cents, or 0.5 percent, from their previous close.
Brent crude futures were at $65.88 per barrel, down 33 cents, or 0.5 percent.
Monday's price falls in part reversed increases last Friday, which came on concerns over tensions in the Middle East.
"Despite all the bearish U.S. shale supply headlines, oil prices remain firm as ... the odds that the U.S. will pull out of the Iran nuclear agreement continue to run very high," said Stephen Innes, head of trading for Asia-Pacific at futures brokerage OANDA.
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.
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 by the Middle East dominated Organization of the Petroleum Exporting Countries (OPEC) to curb supplies and bolster prices.
Many analysts expect global oil markets to flip from slight undersupply in 2017 and early this year into oversupply later in 2018.
One risk to supplies, however, is Venezuela.
"Concerns that Venezuelan output is on the verge of collapse continue to swirl around the market," ANZ bank said.
The International Energy Agency said last week that Venezuela, where an economic crisis has cut oil production by 50 percent in two years to 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 Richard Pullin and Joseph Radford)