The speculation came after Unai Emery was sacked from Arsenal after 18 months in charge following a run of seven games without a win.
The Foxes find themselves in second position in the Premier League after picking up 29 points from their opening 13 games - more points than they had in the 2015/16 at the same stage of the season.
'I’ve loved every minute here'
When asked about Unai Emery's sacking, Rodgers stated: "He’s proven over a number of years that he’s a talented coach. I feel for him and sympathise with him.
"I feel the same as I always have done. If you’re doing ok, people want to take you away from something you enjoy."
He added: "Whenever jobs come up, there’s always a number of names. We’re at the beginning of something here. I’ve loved every minute here. We feel we are beginning something exciting here.
"Arsenal is a great club, and I’m sure they have an idea of who they want to bring in."
'I came to Leicester with a purpose'
Speaking further about the speculation about his future, Rodgers reassured fans, stating: "I’m very happy. If you trace my career, I’m always in a job for a few years.
"I came to Leicester with a purpose. I’m so happy here, every part of my life, the players especially.
"At this moment in time, I’m getting great joy in seeing the work on the training ground move to the pitch."