Reports in France claim the 2015/16 PFA Player of the Year is on the verge of leaving the Premier League champions to join rivals Arsenal.
Mahrez scored an excellent goal on Saturday as Leicester drew 1-1 with Celtic in the pre-season International Champions Cup at Celtic Park.
Ranieri says his star player remains "happy" and warned the winger that he could find himself on the bench if he decides to join a bigger club.
"Did you see how Riyad is happy? He played so well, worked so hard for the team, he scored a goal and he stays with us," the Italian said.
"It is very important [to keep him] he is one of our stars.
"Other teams maybe have more stars and maybe he doesn't play at the same level, because maybe is on the bench so it is much better he stays with me."
When asked by Sky Sports if he is confident Mahrez would stay, he replied: "I am always confident. Of course, we know that there are other teams that want to win the Premier League, we our just to defend."