Icardi, who only recently signed a new long-term contract at San Siro, is currently under fire from Inter's ultra group Curva Nord, who have taken exception to comments made from the forward in his autobiography.
Curva Nord issued a statement ordering Inter to take the captain's armband away from Icardi, amid claims that he threatened to bring over criminals from his native Argentina to kill the club's ultra group.
Before Sunday's Serie A clash with Cagliari, which Inter lost 2-1, Zanetti has revealed that the club will "unfortunately" be taking action against Icardi.
"Will we be taking action? Unfortunately yes. The fans are the most important element, they accompany us and we all should respect them," Zanetti told Mediaset Premium. "We cannot accept this behaviour from a person who works for our club. After this match we will talk things over and see what happens.
"Will we remove the captain's armband? We'll talk about it later, but this is an issue not just for Icardi, but for everyone who is part of the club. Behaviour must be in line with the values of the history of Inter. Nobody expected this outburst - as you know we are concentrated on other objectives - but Inter stands beyond any individual person and we really could've done without this incident ahead of an important match.
"Unfortunately social media can be used in a negative fashion and we must safeguard the club. Every individual who works for Inter must be very careful on social media - he can write anything on his private life, but cannot damage the history of this club."
The spat started when Icardi criticised a member of Inter's ultra group regarding an incident when he tried to give a young fan his shirt after a game.