Bonucci succeeds Giorgio Chiellini, who leave at the end of the season.
Speaking in a press conference ahead of Monday's clash Lazio, Allegri confirmed Bonucci would wear the armband from next season.
The coach also stated: "The team's personality has grown throughout this season.
“The defeat to Inter in the league stopped us in our tracks.
“In Wednesday's match, we were unable to bring him the Coppa Italia. When you don't win, you don't need to talk but focus on what needs to be done next season."
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