The Foxes manager has been touted as a potential candidate to replace Unai Emery at The Emirates after the Spaniard was sacked by the Gunners last week.
Arsenal have appointed club legend Freddie Ljungberg as their interim head coach but the north London side are on the hunt to find a new long-term permanent manager.
Rodgers has been widely hailed for transforming Leicester into title challengers in eight months despite taking over at the King Power Stadium back in February.
Leicester are ahead of Manchester City and Chelsea FC in the Premier League title race, while the Foxes narrowed Liverpool FC’s lead to eight points after their 2-1 win over Everton on Sunday.
Former Liverpool FC defender Carragher doesn’t believe Rodgers should consider taking up the reins of Arsenal given his momentum at Leicester right now.
“I think the position he is in, there is no doubt you want to get right to the top,” Carragher told Sky Sports.
“Leicester are second now but they aren’t one of the top clubs in the league.
“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.
“If he moved now, he would get that tag of jumping ship very quickly. He has got to look at the next 18 months and think he is going to do something with Leicester.
“He will at one stage want to go to one of those big clubs.”
Leicester have won their last six Premier League games to cement their place in the top four.
The Foxes will have the opportunity to cut Liverpool FC’s lead at the top of the table when the Reds make the trip to the King Power Stadium on Boxing Day.
Rodgers has previously managed Swansea City, Liverpool FC and Celtic.
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