Hamburg says both sides reached agreement on the deal Monday morning for the U.S. international but has not disclosed a loan fee.
The 19-year-old, who was born in Florida to a German mother and an American father, is yet to make his Bundesliga debut. The attacking midfielder appeared as a substitute in Bayern's Champions League group game at CSKA Moscow last November.
He has a contract through June 2017 with Bayern, which he joined in 2009 from amateur side SG Hausham.
Green has three appearances for the U.S. after appearing for Germany at youth level. He scored in extra time in the Americans' 2-1 round of 16 loss to Belgium at the World Cup.