* Euro zone manufacturing shrinks, outlook dims
* U.S. crude, products stocks rose last week-API
* Coming up: EIA oil data at 1430 GMT
By Jessica Jaganathan
SINGAPORE, July 25 (Reuters) - Brent remained steady at $103 a barrel on Wednesday as worries about oil demand from the troubled euro zone and the United States offset any gains to oil prices sparked by concerns about supply from the Middle East.
The euro zone's private sector shrank for a sixth month in July as manufacturing output nosedived, notably in the core countries of Germany and France, adding to the likelihood that the bloc will slump back into recession, business surveys showed on Tuesday.
Also weighing on prices was further evidence that the economy of the world's biggest oil consumer was slowing: data showed U.S. manufacturing activity in July expanded at its slowest pace since late 2010.
An improvement in China's manufacturing sector, and the conflict in Syria as well as Iran's tensions with the West over its nuclear programme, kept oil prices supported.
Investors anticipate the Chinese government may take more action to boost the economy of the world's No. 2 oil consumer.
Brent crude inched up four cents to $103.46 a barrel by 0257 GMT while U.S. crude fell 13 cents to $88.37.
"After the sharp drop on Monday, Brent has been trading within a narrow range as there are still some concerns on the euro zone economy, so there's some room for it to decline," said Ken Hasegawa, a commodity sales manager at Newedge Japan.
"The Chinese PMI data had a small impact but there is still a lot of uncertainty on the euro situation...Brent will probably stay in the range of $98-$108 over the next few months."
Spain on Tuesday paid the second highest yield on short-term debt since the euro's birth, which investors interpreted as another sign the country will likely need a full sovereign bailout.
The euro fell to a two-year low against the U.S. dollar and the dollar index strengthened, helping to check dollar-denominated oil's gains.
An unexpected rise in U.S. crude oil stocks also affected investor sentiment. U.S. crude stocks rose 1.3 million barrels last week, the industry group American Petroleum Institute said in its weekly report on Tuesday, surprising analysts who expected a decline.
U.S. crude oil inventories had been forecast to drop 700,000 barrels, with distillate stocks having risen 1.1 million barrels and gasoline stocks slipping 600,000 barrels, a Reuters poll taken ahead of weekly inventory reports showed.
Supporting prices was the escalating violence in Syria, as well as the tensions between Iran and the West over Tehran's nuclear programme.
Syria sent thousands of troops surging towards Aleppo in the early hours of Wednesday, where its forces have been pounding rebel fighters from the air, engulfing the country's largest city in total warfare to put down a revolt. (Editing by Miral Fahmy)