Apple Inc.’s efforts to line up a second supplier for its high-end smartphone screens—and reduce its dependence on Samsung Electronics Co.—have hit a hurdle, according to people familiar with the matter.

South Korea’s LG Display Co. hopes to provide organic light-emitting diode displays for iPhones slated for release this fall, the people said. However, manufacturing problems have caused LG to fall behind the schedule that many suppliers follow to begin mass production for new iPhone models, which usually starts around July,...