Leicester have refused to respond to reports that former England manager Roy Hodgson has been approached about their managerial vacancy.
The reigning Premier League champions sacked title-winning boss Claudio Ranieri last Thursday.
In the first game since Ranieri's departure - under caretaker boss Craig Shakespeare - the Foxes beat Liverpool 3-1 on Monday night to climb out of the relegation zone.
Shakespeare, who was Ranieri's assistant, could be a candidate to take the role until the end of the season or on a permanent basis.
Hodgson was England manager until suffering the ignominy of an exit to Iceland at Euro 2016. His contract with the Football Association was not renewed.
Hodgson, who has stated his desire to return to management, is the latest big name to be linked with Leicester.
Guus Hiddink, who has twice served as Chelsea caretaker boss, and former Manchester City manager Roberto Mancini have also been rumoured to be under consideration to succeed Ranieri. Italian Mancini had a short spell as a player with the Foxes.
Nigel Pearson, whom Ranieri succeeded as boss in July 2015, has also been linked with a return.
However, the circumstances of Pearson's departure - his son James was dismissed as a player for his role in a racist sex tape in the Leicester owners' homeland of Thailand - would likely discount that prospect.