Apple Inc. is planning to build a significant new business in original television shows and movies, according to people familiar with the matter, a move that could make it a bigger player in Hollywood and offset slowing sales of iPhones and iPads.

These people said the programming would be available to subscribers of Apple’s $10-a-month streaming-music service, which has struggled to catch up to the larger Spotify AB. Apple Music already includes a limited number of documentary-style segments on musicians, but nothing like...