The defender has been capped at Under-16 and Under-17 level for Scotland and featured in Celtic's pre-season games, but has opted to continue his career with the Bundesliga giants.
Morrison will join the German champions' development team.
Bayern Campus boss Jochen Sauer said: "Liam had many offers from well-known European clubs."
"We are very glad that he has decided on us. He is a robust, physically strong defender who also brings a lot of quality in his play."