* Brent, WTI at highest since start of oil slump
* Physical Brent above $70 per barrel
* Asian crude grades average above $70 per barrel
* OPEC cuts, global market rally have pushed up prices
* But analysts warn of overheating (Updates prices and quotes, adds physical Brent above $70)
LONDON, Jan 10 (Reuters) - Crude oil prices hit new multi-year highs on Wednesday as OPEC-led production cuts and healthy demand helped to balance the market, but analysts warned of possible overheating.
A broad, global market rally, including stocks, has also been fueling investment into crude oil futures.
U.S. West Texas Intermediate (WTI) crude futures were at $63.60 a barrel, up 64 cents, at 1259 GMT. Earlier prices rose to $63.67, the highest since Dec. 9, 2014.
Brent crude futures were at $69.33 a barrel, 51 cents above their last close. Brent earlier hit $69.37, the highest since May 2015.
"We're still drawing U.S. stocks and that continues to support a very positive sentiment," said Olivier Jakob, managing director of PetroMatrix.
He noted that physical Brent <BFO-> was above $70 per barrel - a psychologically important level.
"It will trigger some increased discussion within OPEC," Jakob said.
The Organization of the Petroleum Exporting Countries, together with Russia and a group of other producers, last November extended an output-cutting deal to cover all of 2018.
The cuts were aimed at reducing a global supply overhang that had dogged oil markets since 2014.
But some in the producer group fear current price gains could prompt shale companies to flood the market.
U.S. crude oil production <C-OUT-T-EIA> is expected to hit 10 million barrels per day (bpd) next month, leaving only Russia and Saudi Arabia at higher levels.
American Petroleum Institute data on Tuesday showed U.S. crude inventories falling by 11.2 million barrels in the week to Jan. 5.
Additionally, the U.S. Energy Information Administration (EIA) raised its 2018 world oil demand growth forecast by 100,000 bpd from its previous estimate.
Official EIA stocks data is due at 1530 GMT on Wednesday.
Oil prices have risen more than 13 percent since early December, and there are indications of overheating. Analysts warned that the market is ignoring U.S. production increases at its peril.
The average price for Asian crude grades has risen to $70.62, Thomson Reuters Eikon data showed, which has pressured the region's refinery margins.
"Selective perception is the reason why the market is completely ignoring this just now," Commerzbank analyst Carsten Fritsch said of rising U.S. production. "Attention is paid only to news that tallies with the picture of rising prices."
(Additional reporting by Henning Gloystein and Roslan Khasawneh; in Singapore; Editing by Dale Hudson)