The 40-year-old, who joined Inter in 1995 and has made over 800 appearances for the club, returned after seven months out of action when he was introduced as an 82nd-minute substitute for Saphir Taider at the weekend as Walter Mazzarri's men picked up a 2-0 win at San Siro.
Zanetti's current deal is set to expire at the end of the season, but he would like to extend his stay by another year.
"I would like to remain and give my contribution to this great family that Inter is," he told the club's official website. "The game against Livorno was a game full of emotion. After so much time away from the pitch, I dreamt of such a night."
Inter's victory maintained its grasp on fourth position in Serie A, seven points adrift of leaders Roma. And Zanetti thinks the Inter squad is capable of competing for the title.
"The Italian league is difficult and there are many teams that compete for the same aims as we do," he said. "But we are working well and we want to fight for something important."