Carragher says Rodgers faces developing a 'ship-jumping' tag if he leaves Leicester for Arsenal.
Carragher said, "There is no doubt, if you are a player or a manager, you want to get right to the top.
"At this moment Leicester are second and they won the title a few years ago. But we would be lying if we said they were one of the top six clubs.
"I don't think it is right if he is even thinking about moving to Arsenal right now. He got a lot of stick when he left Celtic before the end of last season, which was a tough decision for him - because this job was a job everyone wanted.
"If he moved now, he would get that tag of jumping ship very quickly - and he probably would deserve it.
"He has got to look at the next 18 months and think he is going to do something with Leicester.
"After those 18 months, maybe a Tottenham or Arsenal, he will at one stage want to go to one of those big clubs."