CHENGDU, China—China’s campaign to become a global power in advanced manufacturing is playing out at a high-tech factory in Sichuan province, where engineers are aiming to supply display screens for Apple Inc.’s top-tier iPhones.

BOE Technology Group Co. plants already make display screens for Apple’s iPads and MacBook computers, and the company is also the world’s top producer of large liquid crystal screens. Now it is seeking to supply Apple with advanced organic light-emitting diode, or OLED, smartphone screens, according...