Apple Music should also get a lift from the recent release of Apple's HomePod smart speaker, which is programmed to play songs using Apple's music streaming platform.
Apple's subscriber numbers may have already surged ahead of Spotify in the U.S. when including people who are still in free trial periods, the Journal said.
Unlike Spotify, Apple Music doesn't offer a free tier of service, but its subscribers are automatically entered into a free trial when they subscribe.
As a result, Apple Music has three to four times the number of free trial users as Spotify.
Despite Apple Music's rising popularity, Spotify has long held the No. 1 spot as the most popular streaming service in the U.S. and around the world.
Spotify has nearly twice the number of Apple Music's subscribers on a global basis, the Journal noted.
The Swedish streaming platform announced earlier this month that it hit 70 million paid global subscribers, compared to Apple Music, which counts 36 million paid listeners worldwide.
Spotify has the benefit of being an early player in the music streaming market, having launched in 2008.
Apple Music, which launched in 2015, was a comparably late entrant.
Other players, such as Google's YouTube Red, Amazon Music and Pandora, have also tried to carve out a share of the market.
YouTube is widely expected to launch a paid music streaming service, internally referred to as Remix, in March.
The service would compete head to head with Spotify and Apple Music.
The report could potentially put a damper on Spotify's long-awaited IPO.
Last December, Spotify filed confidential initial public offering documents with the Securities and Exchange Commission.
Unlike other high-profile IPOs, Spotify is seeking a direct listing on the New York Stock Exchange.
Direct listings avoid some of the procedures and costs associated with typical IPOs.