The 26-year-old, who joined Partenopei for £8.5m in 2009, has impressed during his loan spell with Atalanta BC this season.
Speaking to TMW, Giovanni Bia said that the Rossoneri could make a move for the 26-year-old in the January transfer window.
"He is a player who Milan like," said Bia. "There was contact between clubs in the summer, but then Cigarini returned to Bergamo on loan.
"A January move? It all depends on the desire of Milan and also on whether Atalanta could find a suitable replacement for him.
"After all, Cigarini is not an important player for Atalanta, he is fundamental for them."
Cigarini also had a loan spell with Sevilla last season.