The 16-year-old has agreed to join his sibling in Munich next season and is expected to play in Bayern's Under-17 side after completing his free transfer.
Bayern's youth director Michael Tarnat confirmed the Bundesliga champions had acquired the services of the young goalkeeper.
"Felix has left Dortmund for Bayern, that is correct," he told Sport Bild.
The younger Gotze not only follows his brother - who made the switch from Dortmund last year - to the Allianz Arena, but also striker Robert Lewandowski, who was unveiled in Bavaria last week.
Mario Gotze scored 15 goals in 44 appearances in his first season at Bayern, and the 22-year-old etched his name into German footballing folklore last Sunday by scoring the winning goal for his country in the FIFA World Cup final against Argentina.
