* U.S. dollar has fallen to 2014 lows this week
* Weaker dollar supports general fuel demand
* But demand outlook weakens ahead of refinery maintenance season
* Rising U.S. output undermines OPEC/Russia supply cuts (Updates prices)
LONDON, Jan 26 (Reuters) - The oil rally paused for breath on Friday after hitting fresh three-year highs in the previous session, but weakness in the dollar continued to underpin prices.
Brent crude futures stood at $70.22 per barrel at 1250 GMT, 20 cents below their last close. On Thursday, the contract climbed to as high as $71.28 per barrel, its highest since 2014.
U.S. West Texas Intermediate (WTI) crude futures were steady at $65.51 a barrel. On Thursday, they also reached their highest since December 2014, at $66.66 per barrel.
Both contracts were set for weekly gains after support from a weakening dollar, which on Friday hit new three-year lows against a basket of other leading currencies.
"For as long as the U.S. dollar remains on the defensive, no more pronounced price fall on the oil market is likely to ensue," Commerzbank analyst Carsten Fritsch said in a note.
As oil is traded in dollars, swings in the greenback can impact oil demand as they affect the price of fuel purchases for countries using other currencies.
Still, crude prices were capped by seasonally weakening demand.
Georgi Slavov, head of research at commodities brokerage Marex Spectron, said despite a generally healthy outlook, there were short-term oil demand headwinds due to the coming end of winter in the northern hemisphere.
Many refiners shut down after winter for maintenance, resulting in lower orders for crude, their most important feedstock.
"Demand is starting to weaken as ... refining capacity was taken out of the market," Slavov said.
This is reflecting in oil inventories. U.S. bank Morgan Stanley noted that global oil stocks built up overall in the week ending Jan. 19.
On the supply side, U.S. oil production <C-OUT-T-EIA> is expected to hit 10 million bpd soon, putting it on a par with top exporter Saudi Arabia.
Output has grown by more than 17 percent since mid-2016. Only Russia produces more, averaging 10.98 million bpd in 2017.
Rising U.S. output threatens to undermine the supply restraint led by the Organization of the Petroleum Exporting Countries (OPEC) and Russia, aimed at propping up prices.
The cuts, coupled with demand growth, have contributed to a near 60 percent rise in oil prices since mid-2017 as excess crude inventories have been drawn down.
(Additional reporting by Henning Gloystein in Singapore; Editing by Susan Fenton)