The 47-year-old Herrlich, who had taken over in 2017, left despite the team's 3-1 win on Saturday against Hertha Berlin that lifted them to ninth place.
Bosz had briefly coached Borussia Dortmund last year after spells at several Dutch clubs, including Ajax Amsterdam, and a stint in Israel in 2016.
He will take over on Jan. 4 when the team meet up after the winter break ahead of the season restart on Jan. 18, and has signed a contract to 2020.
"Heiko Herrlich gave our team important impulses last year after a very difficult season prior to that and led us back to the international scene," Leverkusen sports director Rudi Voller said in a statement.
"Until recently we were determined to stick with him as head coach and turn the corner.
"But unfortunately we cannot deny the fact that there is currently a stagnation in the team's development. We are in an unsatisfactory position at the halfway mark and we considered a change of coach as necessary."
The 55-year-old Bosz led Ajax to the 2017 Europa League final, which they lost to Manchester United, before joining Dortmund. But a bad run until December 2017 led to his departure that month.
(Reporting by Karolos Grohmann; editing by Clare Fallon)
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