The Italian champions have called a news conference for 1300 GMT at which Allegri and the club's sporting director Giuseppe Marotta are expected to address the media about Conte's departure and his replacement.
Italian media report that Allegri has signed a two-year deal that will earn him two million euros ($2.71 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 disastrous World Cup campaign in which his side were eliminated at 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.
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.
In the aftermath of Conte's departure, Juve's deal with Juan Iturbe looks to have fallen through, with the Hellas Verona attacker reported to be moving to title rivals AS Roma in a 30 million euro deal.
(Editing by John O'Brien)
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