Unai Emery was relieved of his managerial duties at Arsenal on Friday following Thursday’s 2-1 defeat to Eintracht Frankfurt in the Europa League.
The Spaniard also failed to win any of his last seven games in all competitions.
Meanwhile, Rodgers has guided Leicester to second place in the Premier League with nine wins from their opening 13 games.
And the former Liverpool boss has distanced himself from the vacancy at the Emirates.
He said: “It’s natural that there are links. If you are doing okay people want to take you away from something you enjoy. If you’re not, it doesn’t matter.
“We’re at the beginning of something here. My relationship with the people here and the players is very strong. We think we are starting something exciting.
“Arsenal is a fantastic club. One of the greats in this country. If the people at Arsenal have moved Unai on, I’m sure they have an idea of who they want to bring in.
“There’s disappointment whenever a coach loses his job. Unai has proved himself over a number of years. I have sympathy for him.”
Former Arsenal winger Perry Groves believes Rodgers would jump at the chance to succeed Emery.
The 54-year-old, who played for the Gunners between 1986–1992, told talkSPORT: “They need to go with experience.
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“They need someone with Premier League experience with a clear vision and a clear direction. There are three candidates, for me.
“Brendan Rodgers – you would go all out to get him no matter how much he cost, if you could.