There have been rumors for a couple of years now of an Amazon Phone presumably running some variant of their forked Kindle Android OS.
But the latest rumor is pretty extreme. Not a low cost phone. But a free ($0.00) phone and without requiring a contract.
Honestly I’m not sure they could pull that off. They haven’t managed to do it with the Kindle Fire or the Kindle after all, though both are heavily subsidized.
Similarly Google who have been pretty aggressive in their off contract Nexus phone pricing haven’t got that down to zero yet.