The Northern Irishman is reportedly Arsenal's preferred choice to take over from Unai Emery, who was sacked on Friday after a torrid start to the season.
Rodgers only assumed the helm at the King Power Stadium in February but has wasted little time stamping his authority on the club.
He has masterminded a stunning start to the season, guiding the second-place Foxes to 10 wins in their opening 14 Premier League games.
Speaking after his side's last-gasp win over Everton on Sunday, Rodgers hinted he has a release clause in his contract.
However, he has now moved to silence suggestions that was an indication of his interest in the job at the Emirates Stadium, claiming he plans to be at Leicester for the long haul.
"I have a contract here until 2022 and thus far – and I know managers are losing their jobs – but the club have not indicated to me that they are going to sack me," said Rodgers.
"I think they're pretty happy with how we are working and like I said, I'm very happy, and I have no reason to look elsewhere.
"I think logic would tell you – I know in your job 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 that's got so much potential and want to improve and develop.
"I was asked a question about my contract, talking about clauses and stuff like that, and I gave an honest answer which was based on most managers probably have something in their contract. And by all accounts that opened the door for something.
"I repeat, I'm very happy here. I think there is a lot for us to develop over the next few seasons. We are moving to a new training facility.
"I respect that there'll always be this speculation around jobs and there is probably for a number of managers.
"For me, I repeat, I'm happy here, very happy and have great relations with all the people and I feel I want to continue with that. Take from that what you want."
Rodgers will hope to guide his side to a seventh consecutive Premier League win on Wednesday when bottom club Watford are the visitors.