The Daily Star says the Northern Irishman is amongst the candidates for the position but finds himself in third place with Leicester following a fine start to the season.
However Rodgers privately regards the Arsenal role as his dream job, sources close to the manager have claimed.
It's believed that the former Liverpool boss is currently caught in a dilemma, especially with his Foxes side doing so well at the moment.
Rodgers is among the top candidates to replace Unai Emery, who was relieved of his duties on Friday morning.
He controversially left Celtic mid-season to take the job earlier this year and has transformed Leicester into serious top four contenders.