Rodgers has returned to Parkhead four years after leaving for Leicester City to replace Ange Postecoglou, who has joined Tottenham Hotspurs.
The Northern Irishman has been given a three-year contract for his second coming.
Rodgers won seven trophies during his first spell as Celtic manager between 2016 and 2019, including the domestic treble twice.
He left Celtic top of the Scottish league table midway through the 2018/2019 season to take charge of Leicester City.
His decision to leave when Leicester came calling irked a lot of Celtic fans but he is determined to win them over once again.
"As a Celtic supporter, I understood what it meant. So that was my regret, that I hurt Celtic supporters and it's a big part of coming back," Rodgers told a media conference.
"Hopefully in time, I can give you that feeling I gave you the first time. I don't expect anything, but if I get that support, great.
"For those who doubt, I've had that all my career so I will continue to work hard and hopefully a team that plays with the commitment people will enjoy. It will just take time.
"I've signed for three years and I guarantee I'll be here for three years unless I get emptied before that, as they say up here."
