SHANGHAI, June 7 (Reuters) - China stocks fell on Friday, despite growth in the country's exports for a second month in May and at a faster pace, after a report that U.S. lawmakers pushed to ban Chinese battery firms with ties to Ford and Volkswagen from exporting to the U.S.

In contrast, other Asian markets rose ahead of a crucial U.S. jobs report that should provide clues on the timing of Federal Reserve interest rate cuts.

Outbound shipments from China grew 7.6% year-on-year in value in May, customs data showed on Friday, providing some relief to the economy as it battles to mount a durable recovery.

Denting sentiment, the Wall Street Journal reported on Thursday that Chinese battery companies with ties to Ford and Volkswagen should be banned from shipping goods to the U.S., a group of Republican lawmakers said, alleging their supply chains use forced labor. The reported weighed down the stock market.

** At the midday break, the Shanghai Composite index was down 0.23% at 3,041.79 points.

** China's blue-chip CSI300 index was down 0.72%, with its financial sector sub-index lower by 0.04%, the consumer staples sector down 1.06%, the real estate index up 1.1% and the healthcare sub-index down 0.57%.

** Chinese H-shares listed in Hong Kong fell 0.55% to 6,519.23, while the Hang Seng Index was down 0.42% at 18,399.04.

** The smaller Shenzhen index was down 0.43%, the start-up board ChiNext Composite index was weaker by 2.26% and Shanghai's tech-focused STAR50 index was down 0.88%.

** China's Contemporary Amperex Technology (CATL), a partner to Ford, and Gotion High Tech, a battery company partially owned by Volkswagen, should be added immediately to an import ban list, the WSJ said.

** Shares of CATL slumped 6.8% and Gotion fell 2.6%.

** Around the region, MSCI's Asia ex-Japan stock index was firmer by 0.12% while Japan's Nikkei index was down 0.21%.

(Reporting by Shanghai Newsroom; Editing by Sonia Cheema)