The Burnley manager, who oversaw his team's relegation to the Championship this term, is on Bayern's radar.
The Bundesliga giants have struggled to hire a coach to replace Thomas Tuchel for next season and beyond.
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The Bavarians are said to want a manager on a one-year contract, as they have eyes on Xabi Alonso for the 2025/2026 season.
Fabrizio Romano claims that Kompany did hold talks with Bayern, but there is no agreement as yet.
There have also been suggestions that Brentford boss Thomas Frank may be a target for Bayern.