Montolivo, signed only last summer from Fiorentina, was chosen ahead of long-serving goalkeeper Christian Abbiati for the role left vacant by the departure of Massimo Ambrosini.
The club's vice-president Adriano Galliani told ANSA: "(Montolivo) is not for sale and will be the new captain of Milan. I spoke with Christian Abbiati and he told me that for him there is no problem."
Italy international Montolivo has been linked with Sevilla but Galliani ruled out any chance of such a move.
"We won't be selling him to anybody," he said.
"Anyway, apart from Barcelona and Real Madrid, no Spanish club can spend 20m euros for one player."