Leicester were looking for Marsch to come in and push their bid for survival in the Premier League, but the former Leeds United boss pulled out of the running.
The Foxes were looking for an interim solution to guide them through to the end of the season, at which time they would consider their options.
Marsch however wanted guarantees over the future and was looking for a longer contract; and it has been claimed that he was unhappy with the answers Leicester gave to him.
The American weighed up the option of taking over at Leicester and decided it was not for him at present.
It is unclear who Leicester will now turn to, with games running out as they bid to climb out of the relegation zone.
Marsch was sacked by Leeds earlier this year as the Whites worried about the threat of relegation down to the Championship.
Leicester were beaten at home by Bournemouth on Saturday and next up have a trip to lock horns with champions Manchester City.
