Apple Music is on the verge of overtaking Spotify AB in U.S. paid subscribers, a sign that the music-streaming world’s dominant force is facing growing competition ahead of its hotly anticipated public stock offering.

Apple Inc.’s streaming-music service, introduced in June 2015, has been adding subscribers in the U.S. more rapidly than its older Swedish rival—a monthly growth rate of 5% versus 2%—according to people in the music business familiar with figures reported by the two services. Assuming that pace continues, Apple...