The 45-year-old is one of the favourites to take over at Emirates Stadium from Arsene Wenger, who will leave the Gunners at the end of the season after nearly 22 years in charge.
Rodgers accepts speculation is part of football but insists he is happy with life in Scotland.
"There's always speculation, none more so in the game than at this time, whether it's players or managers," he told BT Sport.
"I'm extremely happy, I've got three years left on my contract after this year. The club have been brilliant to me since I've come in, as have the players.
"So, I don't think so much about it, to be honest."
Celtic majority shareholder Dermot Desmond said on Friday that the club would not stand in Rodgers' way if he received an offer from Arsenal.
Carlo Ancelotti, Joachim Low and Patrick Vieira are among the other leading candidates to succeed Wenger.