The Celtic boss sits top of the Foxes’ shortlist with their French boss under increasing pressure following defeat in the FA Cup to fourth-tier Newport County and a 2-1 home reverse to Southampton.
Puel has drawn the ire of the Leicester fans for perceived negative play, while results have been varied in recent weeks. They have defeated Chelsea, Manchester City and Everton but suffered losses to Crystal Palace and Cardiff City.
The former Southampton boss took over at the King Power Stadium in 2017.
Rodgers, who has led Celtic to seven successive domestic trophies, is understood to be keen on a return to the Premier League, according to The Sun.
The Northern Irishman has been linked with a host of clubs south of the border since his move to Celtic. He featured on a shortlist for Aston Villa last year, while he has previously revealed that he turned down a lucrative offer to manage in China.
The Leicester link comes at a time when Rodgers has his biggest challenge on the domestic front. Celtic sit top of the table on goal difference as Rangers, Kilmarnock and Aberdeen are all within three points of the summit, albeit Celtic have a game in hand against St Johnstone.
Rodgers has strengthened in the January transfer market by signing a trio of attacking players as the team are set to fight on three fronts between now and the end of the season.