Former Manchester City boss Mancini had signed for three seasons but left after what appeared to be a difference of opinion about the team.
He guided them to the Champions League round of 16 at the expense of Juventus and victory in the Turkish Cup final.
Prandelli, who will be officially unveiled on Tuesday, resigned as Italy's national team manager after the team failed to get past the group stage at the ongoing World Cup in Brazil.
He had been in charge of the Azzurri since 2010, having previously coached Fiorentina, Roma and Parma.
"Galatasaray football team’s new coach Cesare Prandelli is coming to Istanbul this evening," the club said in a statement in Turkish on its website.
"Tomorrow morning at the Florya Metin Oktay (training) facilities, he will be introduced to the players and staff."
Media reports had also linked former Everton and Manchester United manager David Moyes as a candidate for the post.
(Reporting by Josh Reich; editing by Amlan Chakraborty)
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