Bonucci's move from Juve to Milan was officially completed on Thursday in a deal reportedly worth €42 million
The 30-year-old's exit from Turin came amid speculation of a row with Juve head coach Massimiliano Allegri and Bonucci has done little to deny those rumours.
"Having discussions over the years is normal, as I am someone who is very direct and always speaks the truth," Bonucci – who helped Juve to six successive Serie A titles – told La Gazzetta dello Sport.
It came after Bonucci and Allegri were involved in a heated row in the final minutes of Juve's 4-1 Serie A win over Palermo.
"People thought that was the most striking thing, but in truth it was just the final straw: there had been other situations before that," Bonucci said.
"To give 100 per cent, I need to feel important, something that at Juventus was recently happening only in fits and starts."
Braced for a hostile reception when Milan visit Juventus in 2017-18, Bonucci – who could make his debut in the International Champions Cup clash against Bayern Munich on Saturday – said: "They ought to know that, just like the insults I received when wearing the Bianconeri shirt fired me up, it'll be the same if they jeer me."