* Oil traders concerned U.S. may re-impose sanctions on Iran
* Falling Venezuela output also supports oil markets
* Shaky stock markets, fear of trade war caps gains
* Relentless rise in U.S. crude production also weighs
SINGAPORE, March 20 (Reuters) - Oil prices edged up on Tuesday, lifted by tensions in the Middle East, although rising output in the United States and shaky stock markets put a lid on further gains.
U.S. West Texas Intermediate (WTI) crude futures were at $62.31 a barrel at 0128 GMT, up 25 cents, or 0.4 percent, from their previous close.
Brent crude futures were at $66.26 per barrel, up 21 cents, or 0.3 percent.
Traders pointed to concerns in the Middle East, where the United States may reimpose sanctions on Iran, as well as tensions between Saudi Arabia and Iran.
Worries about Venezuela's tumbling crude production also supported oil markets.
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.
Falls on global share markets helped cap gains. Markets are under pressure from concerns over a possible trade war between the United States and other major economies, as well as from fears of stiffer regulation as Facebook came under fire following reports it allowed improper access to user data.
Also looming over oil markets has been surging U.S. crude oil production <C-OUT-T-EIA>, which 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)