BEIJING—Yum Brands Inc. may be a step closer to moving past public scrutiny over the quality of its chicken in China, as Shanghai authorities issued a warning to the fast-food chain but blamed suppliers for recent problems with alleged high levels of antibiotics.

The Shanghai Food and Drug Administration cautioned Yum, saying the Louisville, Ky., parent company of KFC and Pizza Hut should boost its supplier oversight and increase testing of its food products, according to a report from state-run Xinhua news agency. The Shanghai...