The 46-year-old has been touted as one of the candidates to replace the sacked Unai Emery at the Emirates after the Spaniard was sacked on Friday.
But Rodgers, who only joined Leicester from Scottish champions Celtic in February, reiterated his commitment to the Foxes.
Leicester are second in the Premier League, eight points behind Liverpool, after six straight league wins.
"I have a contract here until 2022 and thus far, and I know managers are losing their jobs, but the club haven't indicated to me that they're going to sack me," said Rodgers. "I think they're pretty happy with how we're working
"And I'm very happy, so I've no need to look elsewhere. And I think logic would tell you -- and I know that sometimes in your world it's not always logical, there's gossip and speculation -- but why would I want to leave Leicester City at this moment in time?
"I'm working with a young group of players with so much potential, that want to improve and want to develop. There's a lot for us to grow and develop over these next few seasons. We're moving to a new training facility."
Leicester's next game is against bottom club Watford on Wednesday.