Wenger confirmed on Friday that he's set to step down from the Emirates Stadium hot seat at the end of this season, but Rodgers told BBC Radio Scotland's Sportsound (h/t Alex Young of the Evening Standard):
"There's nothing really in it. I have always said I'm extremely happy here. I have three years left on my contract. I am loving every minute of being at Celtic from a professional perspective but also my life up here.
"They want to get back towards Champion League football, and when Arsene leaves they will look to move in a different direction, I'm sure.
"It's a fantastic club but, for me, my only focus is on Celtic and doing the very best I can for them."
Rodgers moved to Celtic Park in 2016 and is on the verge of becoming the first manager to win back-to-back Scottish trebles, a feat that's assured if they beat Motherwell in the final of this season's Scottish Cup.
The Northern Irishman moved north after parting ways with Liverpool the year prior, and BT Sport commentator Derek Rae commented on how his reputation hasn't been stunted by the transition:
He's familiar with the Gunners as opponents following his spells at Anfield and Swansea City, whom he got promoted to the Premier League in 2011, one of the biggest achievements in his managerial career.
Rodgers added: "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."
Wenger has a year left on his current Arsenal contract but has decided to end his reign at the 22-year mark, having seen his reputation suffer at the Emirates with deteriorating success in recent campaigns.
Whoever takes over at the north London club faces a mountainous task in attempting to steady the ship.
Marca's Chris Winterburn doesn't view Rodgers as the most suitable candidate:
Arsenal have made their decision on Wenger's future and are sure to deliberate long and hard before appointing a successor, a position for which Rodgers seemed one of the more realistic options.
But with things running in his favour in Glasgow and UEFA Champions League qualifiers guaranteed, it appears he's content and will spurn the chance for a Premier League comeback this summer.