The reigning Premier League champions are in real danger of relegation as they sit just one place and one point above the bottom three after a run of five successive defeats.
The Foxes issued a statement last week backing their Italian manager.
"I don't see how Ranieri can keep his job," Sutton told BBC Radio 5 live.
"Had they finished 12th last season and they were on this run of form everyone would be saying he has to go. Football is a ruthless business. The players are not playing for him.
"I think they are in absolutely enormous trouble."
If Leicester go down, they would be the first defending top-flight champions to be relegated since Manchester City in 1938.
They have not won a Premier League game in 2017 or scored a league goal.
Listen to Sutton on Leicester, Manchester City and Liverpool on the BBC Radio 5 live podcast
Leicester City are taking a "huge risk" keeping Claudio Ranieri as manager, says former Blackburn Rovers striker Chris Sutton.