* US says close to finalizing parts of U.S.-China trade deal -CNBC
* U.S. crude stocks fell unexpectedly last week
* OPEC and allies consider deeper oil cuts
* U.S. drillers cut oil rigs for record 11th month -Baker Hughes
* Weak economic outlook drags on prices (New throughout; updates prices, market activity and comments)
NEW YORK, Oct 25 (Reuters) - Oil prices inched up on Friday and were headed for the strongest weekly gains in more than a month as support from falling U.S. crude stocks, optimism over a U.S.-China trade deal and possible action from OPEC and its allies to extend output cuts outweighed broader economic concerns.
Brent crude gained 20 cents at $61.87 by 2:19 p.m. EDT (1819 GMT) but the global benchmark was set for a weekly gain of 4%, its strongest since Sept. 20. West Texas Intermediate (WTI) crude was up 32 cents at $56.55 and on track for a gain of nearly 5% over the week, its best since June 21.
The weekly performance was underpinned by the surprise drop in U.S. inventories, with crude stocks falling by about 1.7 million barrels last week.
"We're holding our ground after a pretty good up week with the surprise draw in inventories this week," said Phil Flynn, senior energy analyst at Price Futures Group.
Oil also got a boost this week from signs of progress in talks on resolving the U.S.-China trade dispute that has weighed on crude demand. Washington officials on Friday said the United States and China were close to finalizing the first part of a trade deal after months of a tariff war.
"That's given us optimism in the crude market that there's going to be some pretty strong demand," Flynn said.
Yet concerns over weakening economic growth continued to drag on prices.
"Slowing global activity will see demand drop, so the reality is that oil rallies will be limited," said Jeffrey Halley, senior market analyst at OANDA.
"It won't take much to pull the rug out from under oil's feet."
Economists in a Reuters poll said a steeper decline in global economic growth remains more likely than a synchronised recovery, even as multiple central banks dole out rounds of monetary easing.
Providing further price support this week, officials at the Organization of the Petroleum Exporting Countries said extended supply curbs are an option to offset the weaker demand outlook in 2020.
Saudi Arabia, OPEC's de facto leader, wants to focus first on boosting adherence to the group's production-reduction pact with Russia and other non-members, an alliance known as OPEC+, before committing to more cuts, sources told Reuters. The alliance in July renewed the pact to cut output by 1.2 million barrels per day since Jan. 1, until March 2020.
"The energy complex is also deriving some support from slowing in the pace of U.S. crude production gains," Jim Ritterbusch, president of Ritterbusch and Associates, said in a note.
In the United States, energy companies reduced the number of oil rigs operating this week, leading to a record 11-month decline as producers follow through on plans to cut spending on new drilling. The rig count is seen as an indicator of future output.
(Additional reporting by Ahmad Ghaddar in LONDON and Aaron Sheldrick in TOKYO Editing by Marguerita Choy and Chris Reese)