Chinese technology giants have leapfrogged U.S. companies in popularizing mobile payments. Now, they are trying to get people to bypass their smartphones and make payments by simply looking into screens.

Ant Financial Services Group and Tencent Holdings Ltd., rivals that operate, respectively, Alipay and WeChat Pay, China’s two largest mobile-payments networks, are competing for dominance in the next stage of China’s cashless society. Each is racing to install its own branded facial-recognition screens at retail points-of-sale...