The Florida-born winger has spent his entire professional career with the German champions, but is yet to make a Bundesliga appearance.
However, that now looks set to change after Hamburg moved to take the 19-year-old on a temporary deal on Monday.
Despite his lack of domestic experience, Green caught the eye during his brief cameo at the World Cup, scoring America's only goal in a thrilling 2-1 extra-time defeat to Belgium in the last 16.
His arrival will come as a welcome boost to Hamburg, who have endured a difficult start to the Bundesliga season.
After being held to a goalless draw at Cologne, Mirko Slomka's men were thumped 3-0 by Paderborn on home soil on Saturday.