The decision outraged sections of the Scottish champions' support and a banner was unveiled at Celtic's subsequent fixture against Hearts that read: "You traded immortality for mediocrity. Never a Celt. Always a fraud."
In addition, video footage has emerged on social media of fans allegedly singing distasteful chants about Rodgers, prompting a response from Celtic that read: "We have no knowledge of who these individuals are, or any connection they have to Celtic, but one thing is certain, such disgusting and pathetic behaviour would never have any place at our club and of course we condemn this completely."
Asked what he would say to supporters who have abused him following his exit from Celtic Park, Rodgers told the Scottish press: "I'd ask them, 'What the f*** are you doing? Bloody hell, I'm one of you!'
"Okay, I made a career decision. But that doesn't make me any less of a Celt."
While Rodgers had earlier acknowledged he understood fans were hurting, he appears to have been surprised by the level of anger at his departure.
He added: "A fraud? Someone who turned up at half seven every morning of his life? Who was in there until all hours of the night? Who took over 60 games a season, preparing a team, changing a culture and a standard at a club, filling the support again?
"It is sad if it [the reaction to his departure] has gone that way. If it is to this depth, for someone who has given so many happy memories to a lot of people and has left the club in the position they are in with the stability, it is sad. I haven't exactly left a sinking ship.
"I don't want that to be the final measure of my work there. I gave much more than that. For my family, it’s probably hard.
"They are lifelong Celtic supporters and have been taken aback by the speed of everything too. It's really difficult for them."
Rodgers takes charge of his first game in the Leicester dugout when his Foxes side face Watford on Sunday.