* Euro zone businesses bounce back in July
* Barclays sees near-tem oil price correction if recovery slows
* U.S. oil rig count rises for first week since March
* China tells U.S. to shut Chengdu consulate (Adds CFTC data)
By Stephanie Kelly
NEW YORK, July 24 (Reuters) - Oil prices rose on Friday, lifted by some supportive economic data, but tensions between the United States and China limited gains.
Brent crude futures rose 3 cents to settle at $43.34 a barrel. U.S. West Texas Intermediate (WTI) crude futures rose 22 cents to settle at $41.29 a barrel.
For the week, Brent rose 0.5%, while U.S. crude rose 1.7%.
Ahead of the weekend, market participants had their eye on Tropical Storm Hanna, forecast to cross to Baffin Bay, 46 miles (74 km) south of Corpus Christi, Texas, on Saturday afternoon or evening.
So far, energy companies said there have been no evacuations of workers or shutdowns of production from offshore platforms in the northern Gulf of Mexico.
Lifting market sentiment, Euro zone business activity grew in July for the first time since the coronavirus pandemic hit, according to IHS Markit's flash Composite Purchasing Managers' Index (PMI). The index is seen as a good indicator of the bloc's economic health.
"The economic data in Europe was much better than anticipated, which would suggest that demand destruction in recent months because of COVID-19 may not have been as bad as people thought," said Phil Flynn, senior analyst at Price Futures group in Chicago.
Meanwhile, U.S. business activity increased to a six-month high in July. U.S. companies, however, reported a drop in new orders as new COVID-19 cases spiked.
The resurgent pandemic has darkened the U.S. economic outlook. Some states have reinstated restrictions, which should reduce fuel consumption.
The number of Americans filing for unemployment benefits hit 1.416 million last week, unexpectedly rising for the first time in nearly four months.
Oil prices could see a near-term correction if a recovery in fuel demand slows further, especially in the United States, Barclays Commodities Research said.
Still, the bank lowered its oil market surplus forecast for 2020 to an average of 2.5 million barrels per day (bpd) from 3.5 million bpd previously.
The U.S. oil and gas rig count, a indicator of future output, fell by two to an all-time low of 251 in the week to July 24, according to data from energy services firm Baker Hughes Co. However, energy firms added one oil rig in the first weekly increase since March.
Meanwhile, money managers raised their net long U.S. crude futures and options positions in the week to July 21 by 5,430 contracts to 375,193, the U.S. Commodity Futures Trading Commission (CFTC) said on Friday.
Weighing on prices, China ordered the United States to close its consulate in the city of Chengdu, responding to a U.S. demand this week that China close its Houston consulate.
Renewed tensions between the world's top two oil consumers further stoked worries about fuel demand.
"Smooth international trade relations are needed for oil demand to remain uninterrupted on the long term and tensions between the U.S. and China are never a good sign," said Bjornar Tonhaugen, head of oil markets at Rystad Energy. (Reporting by Stephanie Kelly in New York; Additional reporting Shadia Nasralla and Julia Payne in London and Jessica Jaganathan in Singapore; editing by David Gregorio, Jason Neely, Mark Potter and Louise Heavens)