UPDATE 7-Oil rises to 3-wk high on Middle East tensions, Venezuela concerns

* Geopolitical factors support prices

* Relentless rise in U.S. crude output caps gains

* Gasoline futures rise to highest since August 2017

* Coming up: API data to be released 4:30 p.m. EDT (Updates prices, market activity, adds commentary; changes byline, dateline to NEW YORK, previously LONDON)

NEW YORK, March 20 (Reuters) - Oil prices rose to their highest level in three weeks on Tuesday as tension in the Middle East and the possibility of further falls in Venezuelan output helped offset the negative impact of growing U.S. crude production.

Brent crude futures for May delivery rose $1.75 to $67.80 a barrel, a 2.7 percent gain by 12:09 p.m. EDT (1609 GMT), their highest level since late February. U.S. West Texas Intermediate (WTI) crude futures for April delivery rose $1.72 to $63.78 a barrel, a 2.8 percent gain.

The more active May U.S. crude futures rose $1.76 to $63.89 a barrel.

"Geopolitics has come to the fore in today's trading session, not least because the Saudi crown prince is on an official visit to the U.S. where the issue of Iran is expected to be on the agenda," said Abhishek Kumar, senior energy analyst at Interfax Energy's Global Gas Analytics in London.

Saudi Arabia called the 2015 nuclear deal between Iran and world powers a "flawed agreement" on Monday, on the eve of the meeting between Crown Prince Mohammed bin Salman and U.S. President Donald Trump.

Trump has threatened to withdraw the United States from the accord between Tehran and six world powers, raising the prospect of new sanctions that could hurt Iran's oil industry.

Worries about falling production in Venezuela, whose output has been halved since 2005 to below 2 million barrels per day (bpd) <PRODN-VE> due to an economic crisis, also supported oil markets.

The International Energy Agency said last week Venezuela was "vulnerable to an accelerated decline" and that the Latin American country could trigger a renewed drawdown in stocks.

However, increased output in the United States, Canada and Brazil has capped oil price gains. The ramped up production threatens to undermine cuts made by the Organization of the Petroleum Exporting Countries in an effort to draw down a global supply glut.

Appetite for U.S. crude is adding to the headache facing OPEC. A widening discount of WTI to Brent crude makes it more attractive for foreign refiners to process U.S. oil. Brent is the benchmark for several Middle East and other global crudes.

The premium of Brent crude to WTI <WTCLc1-LCOc1> rose above $4 a barrel on Tuesday.

Market participants will look to data from industry group the American Petroleum Institute to provide further indications of U.S. supply. Analysts expect the data, scheduled to be released at 4:30 p.m. EDT (2030 GMT), to show that U.S. crude inventories rose for the fourth straight week.

Gasoline futures on the New York Mercantile Exchange rose 2.7 percent in Tuesday's session to a high of $1.9769 per gallon, their highest since August 2017.

Data from market intelligence firm Genscape showed gasoline inventories in the New York harbor region fell by about 1.1 million barrels last week, traders who saw the data said.

(Additional reporting by Amanda Cooper in London and Henning Gloystein in Singapore Editing by Frances Kerry)