The Bundesliga club has struggled to find a replacement for Thomas Tuchel, whose departure was confirmed last week after being announced in February.
Bayer Leverkusen manager Xabi Alonso, Germany manager Julian Nagelsmann, and Austria coach Ralf Rangnick have all declined the post.
Talks to persuade Tuchel to stay failed to yield a favorable result, leaving Bayern with a problem.
Kompany has established a respectable reputation at Anderlecht and Burnley, though the Clarets’ relegation, following lofty pre-season expectations, has not gone well for the Belgian.
