Japan's Sharp Corp. hasn't started mass producing screens for Apple Inc.'s next iPhone, people with knowledge of the situation said Friday, raising questions about whether the U.S. company will have enough components to meet demand for the new smartphone.

Sharp, a major supplier of liquid crystal displays to Apple, had planned to start shipping iPhone screens by the end of August, the people said, but mass production has been delayed in part by manufacturing difficulties. It remains unclear when the company can start shipping...