“There’s nothing really in it. I have always said I’m extremely happy here,” Rogers told BBC Scotland.
Wenger announced on Friday he will step down at the end of the season after 22 years in charge of the Gunners.
Rodgers has restored his reputation in nearly two years in Scotland after a troubled end to his three-and-a-half years at Liverpool in 2015, leading Celtic to the brink of back-to-back domestic trebles in Scotland.
Celtic majority shareholder Dermot Desmond said he wouldn’t stand in Rodgers’s way should he wish to return to the Premier League.
“I don’t think you can put handcuffs on anybody if they want to go to a club as good as Arsenal,” said Desmond on Friday. “It will be Brendan’s decision and Brendan’s decision only.”
However, Rodgers insisted he intends to see out a four-year contract extension at Parkhead he signed a year ago.
“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,” he said.
“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 finally walked under increasing pressure from the club’s fans for change.
Arsenal haven’t won the Premier League since 2004 and face a second successive season without Champions League football unless they win the Europa League.
“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,” added Rodgers.
“It’s a fantastic club but, for me, my only focus is on Celtic and doing the very best I can for them.”