Reports this week have claimed Daniel Levy attempted to bring Rodgers in as a replacement for the sacked Argentine.
Jose Mourinho was appointed as his successor, and Rodgers insists he was never in the running.
Rodgers said on Thursday: "There is nothing to talk on it. The game is full of gossip and speculation.
"The only thing I will say is I was very happy here at Leicester. I've literally just joined the club in February. Everything has been great since we've been in. We have a project here that we want to develop over the next number of years.
"In this game, what I've learned is that you'll always be linked with jobs and speculation around other jobs, but what's most important is the now, and I'm very happy here, and to develop and work with the club in order to help them improve."