Former England boss Hodgson is in the hunt for Leicester job
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 Man 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.
Foxes defender Danny Simpson has, meanwhile, backed Shakespeare to become the club's permanent manager.
The Foxes host fellow strugglers Hull on Saturday, with caretaker manager Shakespeare aiming to cement his chances of getting the job.
Shakespeare has backing from within the club and the squad, and Simpson reckons the 53-year-old is a strong contender to land the role full-time.
He said: "I can't see why not, but it's the chairman's decision.
"It's up to him and he owns the club, he is passionate about the club. Those decisions are nothing to do with us but, for me, he (Shakespeare) has been really good.
"Since day one he has helped me a lot. He is a top coach, a top guy and he has taken it on naturally. He has kept it simple and told us what he wanted to do, which was simple and basic, and we've done that so let's hope we can carry it on for him.
"He's taken it on well and it must have been tough for him."