The Gunners boss announced on Friday that he would be leaving at the end of the season, after managing the North London club for 22 years.
"There is always a lot of speculation around - I have three years left on my contract here, I am very happy," Rodgers told reporters.
"There is always going to be speculation because with Arsene leaving there has to be names put in the hat. I have a number of years left as a manager. I am 45, I probably look a lot older but, 45, and I'm enjoying my position here. It's a huge privilege for me to manage Celtic. It's a huge club with pressure and that's all I concentrate on.
"Arsenal is a fantastic club, very much similar to Celtic in terms of the traditions and values it has as a club, you can't deny that. It's a fantastic club but, for me, my only focus is on Celtic and doing the very best I can for them."