* U.S. fuel inventories post rise amid ample supply
* Weaker refinery margins to prompt lower crude orders
* Iraq, Iran cut physical crude prices to stay competitive
* But surprise U.S. oil output fall supports crude markets
* U.S. crude stocks drop below 5-year average of 420 mln barrels (Adds likely reason for lower U.S. output, updates prices)
SINGAPORE, Jan 11 (Reuters) - Oil dipped away from three-year highs on Thursday on signs that a 13-percent rally since early December may have run its course, although a surprise drop in U.S. production and lower crude inventories offered prices some support.
U.S. West Texas Intermediate (WTI) crude futures were at $63.46 a barrel at 0753 GMT, 11 cents below their last settlement, though still close to a December-2014 high of $63.67 per barrel reached the previous day.
Brent crude futures were at $69.05 a barrel, 15 cents below their last finish, albeit also still close to the previous day's peak of $69.37 a barrel - the highest level since an intra-day spike in May, 2015.
"In Q1, the balance of risk to Brent lies to the downside, with prices overheating, record net-length built into the futures market and fundamentals set to weaken seasonally," BMI Research said in a note.
The mounting downward pressure on prices is also showing in the physical oil market, where OPEC's No.2 and No.3 producers, Iran and Iraq, this week cut their supply prices to remain competitive with customers.
Oil markets have so far been generally supported by a production cut led by the Organization of the Petroleum Exporting Countries (OPEC) and Russia that started in January last year and is set to last through 2018.
More immediate price support came overnight from the United States, where crude inventories <C-STK-T-EIA> fell almost 5 million barrels in the week to Jan. 5, to 419.5 million barrels.
That's slightly below the five-year average of just over 420 million barrels.
U.S. production fell 290,000 barrels per day to 9.5 million bpd, the EIA said, despite expectations of U.S. output breaking through 10 million bpd.
The drop is, however, expected to only be for the short-term, likely due to the extreme cold in large parts of North America which has shut down some onshore producers.
Despite this, more bearish signals are appearing. Fuel inventories in Asia and the United States remain ample, and in some cases are rising.
U.S. gasoline stocks climbed 4.1 million barrels, EIA data showed, more than expected.
In Asia's oil trading hub Singapore, average refinery profit margins <DUB-SIN-REF> have fallen below $6 per barrel this month, their lowest seasonal level in five years, resulting in lower feedstock crude orders.
With the crude price up by more than 13 percent since early December, some analysts expect a downward price correction following the recent bull-run.
"Markets are getting a bit fatigued, and a healthy correction could be on the cards," said Stephen Innes, head of trading for Asia/Pacific at futures brokerage Oanda in Singapore.
(Reporting by Henning Gloystein; Editing by Richard Pullin and Joseph Radford)