The former AC Milan boss says he has a good relationship with the Italy international after they apparently fell out during their time at San Siro.
Pirlo left AC Milan in the summer of 2011, with the Italy international arguing that the 46-year-old preferred to play the likes of Massimo Ambrosini and Mark van Bommel in midfield, effectively forcing him out.
The influential midfielder eventually went on to join Juventus, which he guided to three consecutive Serie A titles, but Allegri's appointment as Antonio Conte's successor means the two are reunited three years after they last worked together.
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"I have always enjoyed a great relationship with him," Allegri said at a news conference. "Andrea was, and still is, a champion. When he was at Milan with me, he went through that season which wasn't particularly lucky for him considering that it was plagued with injuries. But, in fact, he always played when he was fit.
"Unfortunately, in football sometimes you think that things might go one way but eventually they take an unexpected turn or head in another direction. As I have explained a thousand times, in the end it just happened. The club, Andrea and I met and he decided to join Juventus and now I am lucky enough to link up with him again after three years.
"He has enjoyed some unforgettable years here at Juve and now I consider myself lucky enough to find [myself working with] him once again. He had a great World Cup, played three fantastic seasons, so I am lucky. I've never experienced any problem with Andrea and I can justifiably state that I've never called his qualities into question. Otherwise people could justifiably say that I am a fool."
Pirlo has a contract with Juventus until June 2016.