Rodgers was sacked on Sunday after Leicester dropped into the Premier League's relegation zone following a 2-1 defeat to Crystal Palace a day earlier.
The defeat marked a run of five defeats in six Premier League games.
Rodgers spent four years in charge of the Foxes after being appointed in February 2019 and led them to their first FA Cup trophy in the 2020/2021 season.
Leicester finished fifth in each of Rodgers' first two full seasons in charge and reached the Europa Conference League semi-finals last season.
"It has been a privilege to be your manager," Rodgers said.
"I would like to thank the Leicester City chairman Mr Srivaddhanaprabha and all of the board for giving me the opportunity to manage Leicester City and make history with this great football club over the past four years.
"I am obviously disappointed to depart so close to the end of the season and had every confidence that we would have delivered the results needed to preserve the club's Premier League status. I would like to wish everybody at the club the very best of luck as they seek to continue to deliver on this objective.
"I'm now looking forward to having a chance to reflect and recharge and get ready for the next opportunity."
Foxes midfielder James Maddison paid tribute to Rodgers, writing on Twitter: "Thank you gaffer. A top manager and even more importantly a brilliant and caring man.
"Your impact on my career and life outside of football is something I'll be grateful for forever. Good luck with whatever is next."
Leicester's post-Rodgers era begins with a home game against Aston Villa later on Tuesday.
