The defender has been playing in Europe since signing for PSV in May 2008 and will be introduced by America on Thursday.
"We're losing an experienced player who collected merits for the VfB during the one-and-a-half years that he has been here," Stuttgart's director of sport, Fredi Bobic, said in a statement.
Bobic says that Rodriquez dreams of being Mexico's captain for the 2014 World Cup in Brazil and sees his best chance of attaining that goal by playing in his homeland.
"Maza came and asked us to release him. He wants to captain the Mexican national team at the 2014 World Cup in Brazil and to do that, he has to play regularly," Bobic said. "He feels that the opportunity to do that is the highest in his home country."