Earlier, the on demand service signaled its intentions by hiring the former Top Gear presenters Jeremy Clarkson, Richard Hammond and James May to front a new motoring show.
The newly created role will see Green take responsibility for the European video business that was initially acquired from LoveFilm in 2011. It is subsequently been packaged with the Amazon Prime premium delivery service that also includes Amazon Music.
Green joined BT TV in 2012 having spent four years with Virgin Media. He led the distribution of BT Sport, building it to 5 million households in less than two years, and launched BT TV onto the YouView set-top.
Clarkson had been dismissed from the BBC production following an altercation with a producer. Amazon has previously picked up on on Ripper Street, the crime drama initially dropped by the BBC, and has been developing its own slate of original productions in the United States.