"Allegri is the new manager of Juventus," the Italian champions said on their official Twitter feed.
Italian media report that Allegri has signed a two-year deal that will earn him €2 million (£1.5 million) a season plus bonuses.
Conte made the surprise announcement that he was leaving Juve on Tuesday after winning three straight Serie A titles during his three years in charge.
The former midfielder could take the Italy manager's post left vacant by Cesare Prandelli, who stepped down as national team boss after a terrible World Cup campaign in which his side were eliminated following the group stage.
Allegri was sacked by Milan midway through last season after a dreadful start to the campaign that left them languishing in mid-table, and was replaced by Clarence Seedorf in January after more than a year of speculation surrounding his position.
"It’s an honour for me. Let’s say it’s a bolt from the blue. First and foremost I consider myself fortunate," Allegri told reporters at a press conference on the day of his unveiling.
"I understand the scepticism of the fans. I will win them over with work, respect and professionalism."
His arrival could cause friction with Andrea Pirlo after Allegri sold the gifted playmaker to Juve in 2011 while in charge of Milan, convinced that he could no longer play in his favoured position in front of the defence.
However, Allegri took to Twitter to quash any suggestions that he and Pirlo would not see eye to eye.
Allegri tweet: With @Pirlo_official I got a great relationship, we haven't talked yet, he's a champion; with me he's always played #AllegriAllaJuve
Allegri tweet: Luckily after three great seasons I find again @Pirlo_official, and I never put his talent in discussion #welcomemax
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- Andrea Pirlo