The 57-year-old Frenchman's 16-month reign came to an end after Saturday's 4-1 home defeat to Crystal Palace, Leicester's sixth defeat in seven matches.
During that run the club have been knocked out of the FA Cup by Newport and dropped down to 12th in the Premier League, eight points above the relegation zone.
"I would like to thank everyone that I worked with at Leicester City Football Club," Puel said in a statement released through the League Managers Association on Monday.
"It was a privilege to work with the players, the staff and the management.
"It was my honour to work for the late and much-loved Khun Vichai, who gave me the fantastic opportunity to manage his club and share some great moments with him.
"My mission at Leicester City Football Club ends here but I will continue to follow the performances of the team and wish the club all the best for the future."
Puel is the fifth English top-flight manager to leave his post this season, following Slavisa Jokanovic from Fulham, Mark Hughes (Southampton), Jose Mourinho (Manchester United) and David Wagner (Huddersfield).
The former Premier League champions have endured a traumatic season with the death of owner Vichai Srivaddhanaprabha and four others in a helicopter crash outside the club's King Power Stadium in October.
Celtic manager and former Liverpool boss Brendan Rodgers is reported to be among the frontrunners to replace Puel, with reports suggesting the club may appoint an interim manager until the end of the season.