The Gunners sacked Unai Emery on Friday, leaving assistant Freddie Ljungberg in temporary charge with Rodgers mooted as a potential replacement.
On whether he intended to stay with the Foxes long term, Rodgers told BBC Sport (h/t the Daily Mail's Tom Collomosse):
"Yeah, very much so. I left a club eight and a half months ago and that was a change I made to come to a club with ambition.
"And it is a huge challenge here to break into the top six. I've loved every minute since I've come here. We've got work to do but I'm excited.
"My focus is very much here with Leicester. The club has been first class with me. The project feels very exciting with a lot of development ahead. So my concentration is very much with Leicester."
The coach is thought to have a £14 million compensation clause that interested parties could use to buy him out of his contract at the King Power Stadium.
On that issue, Rodgers had earlier told Sky Sports (h/t Collomosse): "There probably is [one] in most managers' contracts. It's all hypothetical, all these types of situations."
The Northern Irishman left Celtic for the Premier League club in February after Claude Puel had been relieved of his duties.
Leicester have enjoyed a superb season under him so far.
They sit second in the league after 14 matches, eight points behind leaders Liverpool, six ahead of fourth-place Chelsea and 12 clear of Tottenham Hotspur in fifth. Leicester boast the best defensive record in the division with nine goals conceded, and only Manchester City have scored more than their 33.
On Sunday, Rodgers guided the Foxes to a 2-1 win over Everton.
Leicester were behind to Richarlison's 23rd-minute header when Rodgers brought on Kelechi Iheanacho for his first Premier League appearance of the season.
The striker made an instant impact with an assist for Jamie Vardy, before netting a dramatic injury-time winner himself:
Vardy has particularly thrived under Rodgers, having now found the net 22 times in 24 league games under him.
Bleacher Report's Jack Collins hailed the manager for turning things around in Sunday's match:
Sports journalist Richard Martin questioned why he might leave the club for Arsenal at this stage:
Although Arsenal have some star players in Mesut Ozil, Pierre-Emerick Aubameyang and Alexandre Lacazette, the Foxes arguably have the better squad overall.
Leicester have an excellent chance of securing UEFA Champions League football this season and an outside chance of repeating their incredible 2015-16 title win, while Arsenal's best hope of doing the former looks to be by winning the UEFA Europa League.
The Gunners are winless in eight games in all competitions after their 2-2 draw with Norwich City on Sunday, a run stretching back to October 24.
They're eighth in the Premier League, seven points off the top four—the same distance they are from Southampton in 18th—and 13 behind Leicester.