The Foxes are searching for a permanent successor to Nigel Pearson who was sacked amid controversial circumstances earlier this month.
It would be an intriguing move for Schmeichel, who played for United from 1991 to 1999.
The 51-year-old Dane has not managed before and would have to deal with overseeing the development of his son Kasper – also a goalkeeper.
Martin O’Neill was believed to be the club’s preferred candidate last week with some supporters keen to bring back the man who presided over the team’s most sustained periods of stability and success.
O’Neill previously managed Leicester for five years, winning promotion to the Premier League in his first season in 1996, but it is understood the club are yet to approach the Football Association of Ireland with their interest.