Unnamed sources told Billboard, as well as industry blog Hits Daily Double, that Iovine's departure might be timed around the vesting of his stake in Apple in August.
Apple declined to comment to CNBC.
Iovine's departure would come at a key time for Apple, which has staked the release of its smart speaker, HomePod, around its identity as a music technology pioneer, even as Spotify aims to go public.
As a boss at Interscope, Iovine helped Steve Jobs make iTunes a reality, and later joined Apple as part of its acquisition of Beats. That 2014 deal was worth $3 billion, with approximately $400 million of that vesting over time.
But Iovine, who is in his 60s, has openly flouted the business behind music streaming companies, even in his role at Apple Music.