Allegri's return comes after a difficult season for Milan, which ended in eighth place in the league and missed out on European qualification. The team also lost in the Coppa Italia final against Bologna.
Allegri has been brought in to replace Sergio Conceicao, who was sacked after five months in charge. The 57-year-old Italian has a clear objective for the season: to return to the Champions League.
"The goal for this season is to return to the Champions League as a club like AC Milan has to play in Europe," Allegri said in Singapore, where his team will face Arsenal in a pre-season friendly.
Allegri won the Scudetto in his first season at Milan and went on to enjoy a successful spell at Juventus, where he won five straight Serie A titles. He believes that with the recent signing of Luka Modric from Real Madrid, Milan can restore its status among Europe's elite.
Modric, a 39-year-old Ballon d'Or winner, still has plenty to offer, and Allegri will be counting on his wealth of experience and leadership to inspire the team.
"Modric is a proven champion who does not need any introduction. He is a role model not only from the discipline but technical perspective," said Allegri.