Subotic, 28, joined Cologne on a six-month loan from in January after enduring a difficult 2016.
Lothar Matthaus has said "there is no place" for Thomas Muller in Carlo Ancelotti's strongest Bayern Munich XI.
Johansson said: "I'm very pleased to join FC Bayern from this summer, and I am looking forward to the challenge."
The rule stipulates that more than 50 percent of a club must be owned by its members, but there are exceptions for company-owned clubs.
Carlo Ancelotti will enjoy his 1000th game as a manager this weekend, as his Bayern Munich side take on Hamburg.
Wood missed Hamburg's draw with Freiburg last Saturday because of a thigh problem.