This rumor is courtesy of BuzzFeed News, which reports that the news will be announced on-stage during Apple's annual Worldwide Developers Conference (WWDC) on June 5. A different source, one "familiar with the companies' thinking" noted that while the app is expected to go live this summer, that isn't set in stone and could still change.
The addition of the long-missing Amazon Prime service fills a gaping void that forces Apple TV owners to use other streaming boxes to see Prime shows such as Man in the High Castle and Mozart in the Jungle. While Prime Video can be found on many devices, such as Amazon Fire TV sticks, Rokus and gaming consoles such as the Xbox One and PS4, moving that HDMI cable back and forth can get annoying fast.
Apple TVs gaining Amazon Prime Video seems to be only the first half of a deal, though, as the report also notes that Amazon will start selling Apple's set-top box again. No word as to the possibility of Amazon and Google burying the hatchet so that the Google Chromecast could also be sold from the online retail giant.
Prime Video might not be the only new service joining the Apple TV family soon. Apple is currently working on its own original content, such as its app development reality show and Carpool Karaoke episodes.