The Italy international hurt his right knee in Saturday's 2-1 triumph over Catania and will undergo an MRI on Monday to confirm the extent of his injury.
"I am really saddened for Giandomenico because he was playing really well and he is an important player for us," Napoli coach Rafa Benitez said to www.sscnapoli.it.
"He has a knee problem and I hope he can recover as soon as possible."
Mesto has made eight league appearances to help Napoli climb to second in Serie A.
The 31-year-old will miss Wednesday's Champions League Group F game against Marseille as well as the showdown against Juventus in Turin on Sunday.