Bosz, 53, led Ajax to last month's Europa League final, which the Dutch club lost 2-0 to Manchester United.
Tuchel left Dortmund at the end of May after two years in charge.
He led the club to victory in the German Cup last season and helped them qualify for the Champions League with a third-place finish in the Bundesliga.
Bosz replaced Frank de Boer at Ajax at the start of last season and was under contract with the Amsterdam side until 2019.
"I have mixed feelings about this move," said Ajax managing director Edwin van der Sar.
"When we brought Peter to Ajax last summer it was not the intention that the partnership would end after one year.
"We have had a great season, especially in the Europa League. As with our players, the coach also attracts the attention of clubs from Europe's top leagues."
Borussia Dortmund have named Ajax boss Peter Bosz as their new manager on a deal until 2019 following the departure of Thomas Tuchel.