Assistant coach since 2011, Shakespeare is now 5-4 favourite ahead of Dutchman Guus Hiddink, former England manager Roy Hodgson and Nigel Pearson, the man who was succeeded by Ranieri.
"It appears that the shortlist for Leicester manager is down to four with Craig Shakespeare leading the way following Leicester's victory over Liverpool," a spokesman for bookmakers William Hill said on Tuesday.
As job interviews go, Shakespeare could hardly have done better on Monday as champion Leicester produced their best display of the season in the Premier League.
Jamie Vardy scored twice and Danny Drinkwater once as Leicester ended a run of five successive defeats to move out of the relegation zone.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"I have said before that, do I think I'm capable of it? Yes. Does it faze me? No," Shakespeare said.
"But I think it is the powers-that-be, the owners, who decide and do their job, it is their diligence to look. I am sure they will do that."
(Reporting by Martyn Herman; Editing by Alison Williams)
