The 45-year-old has emerged as a rumoured possible candidate to take over at the Emirates Stadium should under-pressure Wenger leave the club at the end of the campaign.
Rodgers, who came close to winning the Premier League title with Liverpool in 2013-14, insists that he will not let the ongoing speculation distract him from guiding Celtic to more major honours this term.
"Arsenal is a great club with a great manager," he told The Times. "People will look at it and see that, [when] I went to Liverpool, they were struggling for five seasons for Champions League football and I was able to get them back. When I came to Celtic, they hadn't been in the Champions League for three seasons and I got them back. So maybe people are thinking, OK, Arsenal have been out of the Champions League and that is maybe the equation for someone like myself.
"However, I'm loving my life up here. I'm not going to be here forever but I'm living in a dream, although dreams always come to an end at some point, that's for sure. I'm always relaxed about speculation but my focus is only on Celtic. It's a fantastic club, Arsenal, but they have a great manager. Unfortunately, the criticism Arsene takes is now the way in the modern game, modern football. That's unfortunate because he has done so much for the British game.
"He came in 20-odd years ago and revolutionised the game. And it's disappointing and sad for another manager to see someone of that quality and who is held in such esteem have to go through that. I love my professional life here, the challenges of managing, the expectations, the scrutiny that comes with Celtic.
"Like I say, I'm living the dream. As for the other stuff, you can't control it so I don't worry about it. It will always be the case at big clubs, especially if your results turn the other way."
Wenger, in charge of Arsenal since 1996, has come under increasing scrutiny following three defeats in the space of a week.