Apple declined to comment.
Apple executives have reached out to media companies to buy exclusive rights to podcasts, according to the report.
Spotify has invested heavily in original podcasts. CEO Daniel Ek said it planned to spend $400 million to $500 million on original podcasts this year, including purchases of Gimlet Media and Anchor, two podcast startups.
Gimlet Media produced a variety of original fiction and non-fiction podcasts, including "Reply All," "Startup" and "Homecoming." Spotify bought it for $230 million, according to Neiman Lab.
Apple's Podcasts app is installed on iPhones by default and is the most used podcast app by far according to most estimates. However, Apple doesn't host or bankroll podcasts. Instead, it hosts a directory of links that enables users to download podcasts from other servers.
"This makes a lot of sense from a strategic perspective," Chartable CEO Dave Zohrob told CNBC. Chartable is a podcast analytics company.
"Spotify has been investing heavily, and has quickly grown to be a very strong 2nd-place listening platform," he continued. "Podcast listeners on Spotify tend to spend more time in-app — meaning they listen to more music and audio overall, deepening engagement with Spotify."
Apple's Podcasts app is still the number one app, he said, but Spotify is catching up fast.
Apple was down less than 1% on Tuesday.