Dortmund, who finished third in the Bundesliga and will compete in next season's Champions League group stage, fired Tuchel three days after winning the German Cup on May 27.
Tuchel had a public row with club bosses over playing a Champions League game only one day after a bombing attack on the team bus in April.
The 53-year-old former Dutch international Bosz steered Ajax to the Europa League final this season where they lost to Manchester United.