The ex-Sao Paulo star says he wants to stay at the San Siro despite City's readiness to throw money at him in the hope that he would swap a Serie A title race for a Premier League survival battle.
A City delegation has been in Milan and reportedly made a bid in the region of £100m, with a weekly wage of £500,000 on offer for the 26-year-old.
However, he told a website owned by Milan owner Silvio Berlusconi: "I want to grow old at Milan.
"My aim is to become at some point in the future the captain of this team."
With Paolo Maldini currently holding the AC Milan skipper's armband, Kaka added: "I know there is a hierarchy and Maldini and (Massimo) Ambrosini are before me, but after..."
City's conduct in the affair has been cheeky after they allegedly told Milan they wanted to discuss a deal to sign out-of-favour keeper Dida, before bringing up the topic of Kaka's price-tag at the meeting.