The tactician replaced Conte at Juventus in 2014 after the Italian guided the club to three consecutive league titles.
He went on to win five straight Serie A trophies, four Coppa Italia titles and reach two Champions League finals during his first spell in Turin.
Conte departed Napoli in May after two seasons, leading the club to first and second-place finishes in Serie A.
"I am inheriting a squad that is certainly accustomed to hard work," Allegri said in his first press conference.
"Conte has proven his worth throughout his career. I'm lucky, this is actually the second time I've inherited Antonio's team - let's hope it's a good omen.
"It will be important to work with seriousness and professionalism, and above all to lay the groundwork for reaching March - as I always say that's when seasons are decided - still in contention for all our objectives: Serie A, the Champions League, the Italian Cup."