* IEA monthly report paints gloomier picture for oil demand
* U.S. Dec shale output to rise for 12th straight month - govt (Updates prices)
LONDON, Nov 14 (Reuters) - Oil declined on Tuesday for a third day running as evidence of rising U.S. output and a gloomier outlook for demand growth in a report from the International Energy Agency (IEA) weighed on prices.
Brent crude futures were down 71 cents at $62.45 a barrel by 1510 GMT, while U.S. West Texas Intermediate (WTI) futures fell 64 cents to $56.12.
Last week, both benchmarks hit their highest levels since 2015, but traders said the market had lost some momentum since then.
The IEA on Tuesday delivered a surprisingly downbeat outlook for oil demand in its monthly market report, showing an expected slowdown in consumption that was at odds with a more bullish view from OPEC on Monday.
The Paris-based IEA cut its oil demand growth forecast by 100,000 barrels per day (bpd) for this year and next, to an estimated 1.5 million bpd in 2017 and 1.3 million bpd in 2018.
The IEA said warmer temperatures could reduce consumption, while sharply rising output from outside the producer group OPEC might mean the global market tilts back into surplus in the first half of 2018.
"You cannot have the same forecast at $60 as you have at $40. You need to address that and the IEA is starting to make that adjustment," Petromatrix strategist Olivier Jakob said.
Traders said they were cautious about betting on further price rises.
"Prices ... are starting to look like a pause or pullback is needed," said Greg McKenna, chief market strategist at futures brokerage AxiTrader.
This sentiment comes in part on the back of rising U.S. oil output <C-OUT-T-EIA>, which has grown by more than 14 percent since mid-2016 to a record 9.62 million bpd.
The U.S. government said on Monday U.S. shale production in December would rise for a 12th consecutive month, increasing by 80,000 bpd.
"The recent price support, namely the tension in the Middle East, has been swept aside as rising rig counts and U.S. shale output (are) in the focus of traders," PVM Oil Associates analyst Tamas Varga said.
Despite the cautious sentiment, traders said oil prices were unlikely to fall far, largely due to supply restrictions led by the Organization of the Petroleum Exporting Countries and Russia, which have helped reduce excess stockpiles. (Additional reporting by Henning Gloystein in Singapore; Editing by Dale Hudson and Greg Mahlich)