Rodgers left his role at the club after Leicester’s 2-1 defeat to Crystal Palace, with the Foxes now in the Premier League relegation zone.
Adam Sadler and Mike Stowell were named as caretaker managers of the club, but fell to defeat at home to Aston Villa in their first match in charge.
And, as reported by The Athletic, if they cannot get a positive result against Bournemouth this weekend, Leicester could move quickly to appoint a new head coach.
The report continues to reveal that Marsch has held talks with the Foxes over their vacancy and is one of the candidates under consideration if they bring in a new manager after the match against Bournemouth.
However, Marsch is not the only candidate that has been spoken to as the Leicester board still hope Sadler and Stowell can get the results to see them through to the summer, when they will search for a permanent manager.
For now, an interim manager is still being weighed up, while a permanent appointment could be made if the board feel that this would give them the best chance of staying up.
Rafa Benitez and Sam Allardyce are believed to be possibilities for an interim rone if they are intereted.