The former AC Milan boss has agreed a two-year deal with the Bianconeri, with official confirmation of his appointment expected soon.
Conte guided the Turin side to three consecutive Serie A titles after taking over in the summer of 2011, but decided to resign on Tuesday over his frustration with the club's transfer policy.
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Juventus has wasted no time in finding a replacement, though, and on Wednesday morning Allegri held successful talks with Bianconeri president Andrea Agnelli where the pair agreed on a two-year deal worth 2 million euros a season plus bonuses.
Official confirmation of the appointment could arrive on Wednesday, with Allegri starting his new job on Thursday.
The Juve hierarchy had already held two phone conversations with the 46-year-old on Tuesday evening to sound him out about taking over the Old Lady.
Allegri has been out of work since he was sacked by AC Milan in January of this year and he sees Juventus as the perfect challenge to get his career back on track.