If true, it would be interesting to see what kind of deal is being worked out, or what other changes are afoot. There are fundamental business model barriers that have kept Amazon video off of the ATV. Simply put, Amazon Prime video is a loss-leader for Amazon. Your prime subscription doesn't pay for the content available through Prime. So from Amazon's perspective, if they put their video app on the ATV, they want you to be able to then purchase or rent other non-prime video content from them.
Apple, on the other hand, requires all in-app purchases of digital content to go through their store, where they take a cut from the purchase or rental price. Amazon isn't interested in doing that, because that cut would be their profit margin or more. You can get around that on an iPhone or iPad, because unlike your ATV, those devices already have browsers, and Apple can't take a piece out of every purchase made through a browser. So it's a slight inconvenience on an iPhone, but Amazon will put up with their customers using a browser to complete a purchase, and then switching back to their video app to watch the content. Using AirPlay to then stream from an iPhone to an ATV is just an extension of that model.
So thus far, because the ATV doesn't have a browser, Amazon has avoided putting a video app on ATV, because they would either have to give away their profits for in-app digital content transactions, or offer an app with no internal store, which would lead too many of their customers to watch their expensive, loss-leader Prime content only to then jump over to Apple's app to purchase and rent video content there. People get all mad at Amazon about this, but neither of those options is tenable for them. Both options involve them spending money on loss-leader Prime content, only to then take the loss while leading their customers right into Apple's bank account.
From Apple's standpoint, I can't imagine that this impasse is really that big of a problem for them. They have their own video content. If they give Amazon some kind of special break on their cut of in-app digital content purchases, it would set a precedent where other vendors would want the same deal. They have no interest in doing that.
The only thing I can think of that would allow Amazon to put their video app on ATV would be that Apple is planning for other reasons to start including a browser on the ATV. That seems unlikely, however, because you kind of need a keyboard to effectively work a browser.