After winning the Serie A title with Milan in 2011, Pirlo moved to Juventus on a free transfer and kick-started the Bianconeri's domination of Italian football, winning four more league trophies before moving to Major League Soccer in 2015.
Bonucci has gone the other way, joining Milan for a fee of more than 40 million euros ($45m) after winning six straight Serie A titles with Juventus.
"I hope to do with Milan what Pirlo did with Juve," Bonucci said in an interview with Gazzetta dello Sport.
Like Pirlo, Bonucci has now played for the three biggest clubs in the country. The Italy defender made his Serie A debut with Inter Milan in May 2006.
Bonucci joined Juventus in 2010 from Bari and was a key part of its rebuilding, becoming Italy's top defender in the process. The 30-year-old Bonucci wants to do the same at Milan, but with even more success.
"I see a lot of parallels," Bonucci said. "In fact, I chose Milan also because in my mind there is the idea of repeating the path I lived at Juventus, where we started from a reconstruction like is the case here, and ended up at the top.
"I want to go even higher here. Within four years I hope to manage to win the Champions League. I want to win everything."
Juventus and Bonucci lost two UEFA Champions League finals in three years.
Milan opens the season at Crotone on Sunday, with Bonucci as captain.
"It was a choice shared with the club, the coach and the squad," Bonucci said. "I can make my charisma and experience available, but it's the team that makes you that way.
"I'm lucky to have learned from great leaders like (Gianluigi) Buffon, (Alessandro) Del Piero and (Giorgio) Chiellini."