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UPDATE 5-Oil prices up on weaker dollar, but supply glut caps gain

* Crude oil rises this week after five weeks of decline

* U.S. output growing, rig count highest in over 3 yrs

* OPEC to consider options at July meeting in Russia

* Weekly U.S. API inventory data out on Tuesday (Updates detail, adds OPEC context)

LONDON, June 27 (Reuters) - Oil prices rose for a fourth consecutive session on Tuesday, boosted by a weaker dollar, but worries over persistent oversupply capped gains.

Brent crude futures, the international benchmark for oil prices, gained 46 cents to $46.29 per barrel by 1356 GMT.

U.S. West Texas Intermediate (WTI) crude futures were up 39 cents at $43.76 per barrel.

The gains mean the market is up slightly so far this week, after spending much of the last month in negative territory.

U.S. crude inventory figures due out later on Tuesday could help determine if the market extends its gains.

The dollar fell 0.1 percent against a basket of six major currencies, before a speech by U.S. Federal Reserve Chair Janet Yellen in London.

"Short-term financial investors also significantly scaled back their net long positions in Brent on the ICE last week ... and find themselves at their lowest level in a year and a half," Commerzbank said in a research note.

"Short positions have soared to a new record high, having increased more than four-fold since the beginning of the year."

The Organization of the Petroleum Exporting Countries and its partners have tried to reduce a global crude glut with production cuts. OPEC nations and 11 other exporters agreed in May to extend cuts of 1.8 million barrels per day (bpd) until March.

Despite the cuts, which started in January, inventories have not fallen as expected, as U.S. producers and others outside the OPEC-led regime have boosted output.

Ian Taylor, head of the world's largest independent oil trader Vitol, told Reuters that Brent prices would stay in a range of $40-$55 a barrel for the next few quarters.

"In the third quarter we should draw (down stocks), but we are unsure about the fourth quarter as U.S. production is likely to have a year-end spurt," Taylor added.

OPEC delegates said the cartel will not rush into making a further cut in oil output or end some countries' exemptions from output limits, although a meeting in Russia next month is likely to consider further steps to support the market.

OPEC members Nigeria and Libya are exempt from the cuts and have raised production substantially while Iran has also been allowed a small increase to recover market share lost due to Western sanctions.

U.S. production has risen about 10 percent since last year to 9.4 million bpd <C-OUT-T-EIA>, with the number of U.S. oil rigs in operation at the highest in more than three years.

Colonial Pipeline Co said demand on its main distillate line fell below capacity for the first time in nearly six years as the U.S. East Coast remained awash in fuel.

Bank of America-Merrill Lynch analysts said demand was not growing quickly enough to absorb extra output, especially since imports in Asia are stuttering. (Additional reporting by Naveen Thukral in Singapore; Editing by Edmund Blair and Adrian Croft)