* U.S. rig count rises back to 800 - Baker Hughes
* Rising drilling activity points to higher crude production
* Middle East tensions prevent further price falls
By Henning Gloystein
SINGAPORE, March 19 (Reuters) - Oil prices fell on Monday as rising drilling activity in the United States pointed to further increases in output, raising concerns about a return of oversupply.
U.S. West Texas Intermediate (WTI) crude futures were at $62.02 a barrel at 0145 GMT, down 32 cents, or 0.5 percent, from their previous close.
Brent crude futures were at $65.87 per barrel, down 34 cents, or 0.5 percent.
Monday's price falls in part reversed increases last Friday, which came on the back of concerns over rising tension 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 in Singapore.
On a simple supply versus demand basis, however, global oil markets are facing the risk of returning into oversupply 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 when 631 rigs were active as energy companies have continued to boost spending since mid-2016 when crude prices began recovering from a two-year crash.
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.
Reporting by Henning Gloystein; editing by Richard Pullin