* U.S. economic data lifts crude on both sides of Atlantic
* Weak data on China's factory output weighed on oil prices early
* Saudi Arabia pumps 10 million bpd in July (Updates prices to settlement)
NEW YORK, Aug 21 (Reuters) - U.S. and Brent crude futures rose on Thursday, lifted by supportive economic data from the United States after a plentiful supply picture and Chinese economic data had earlier pressured prices.
Sales of existing U.S. homes rose to a 10-month high in July and the number of initial jobless claims fell last week, signaling third-quarter strength in the economy.
Separate reports on Thursday showing factory activity in the mid-Atlantic region in August at its highest level since March 2011, and a gauge of future economic activity up solidly last month, added to the hopes for improving demand for oil.
"U.S. crude might have a gotten a little ahead of itself to the downside recently given the strong demand from U.S. refiners and the relative of strength of the U.S. economy, as this morning's data show," said John Kilduff, partner at Again Capital LLC.
The world's top crude oil benchmarks have both fallen more than $10 a barrel since June on a build-up of supply in the Atlantic Basin and continued production from Iraq and Libya despite the risk of supply disruption from the region's conflicts.
U.S. October crude futures rose 51 cents to settle at $93.96 a barrel, reversing earlier losses that sent prices to $92.50, the lowest since Jan. 15.
The U.S. September contract expired on Wednesday at $96.07, up $1.59 on the day and with the premium of the front-month over the nearby contract CLc1-CLc2 reaching $3.12 intraday.
Data showing a large drop in U.S. crude inventories last week and refinery capacity use at a robust 93.4 percent helped spark Wednesday's rally.
Brent October crude on Thursday rose 35 cents to settle at $102.63, up from its session low of $101.21. Brent fell to $101.07 on Tuesday, its lowest since June 26, 2013, the same month it was last traded under $100 a barrel.
A survey of China's factory activity showed that growth in the sector slowed to a three-month low in August, pressuring oil prices early on Thursday and fueling concerns about a soft economy dampening demand for oil in the world's second-largest oil consumer.
Along with worries about demand, increasing supplies from OPEC member nations has eased fears of supply disruptions.
Libya has resumed exports from its largest port, helping lift oil production to its highest in months, while Saudi Arabia raised its output in July to 10 million barrels per day (bpd).
"Supply fears have been quelled by an increased export volume from Libya," said Dorian Lucas, an analyst at energy consultancy Inenco.
As Brent's retreat neared $100, there was talk that OPEC could consider cutting output, although delegates from the producer group have said higher seasonal demand in coming weeks is expected to support the market. (Additional reporting by Robert Gibbons in New York, Christopher Johnson and Henning Gloystein in London Florence Tan in Singapore; Editing by Marguerita Choy)