The former Middlesbrough boss currently leads the Birmingham City betting.
The former midfielder is being chased by both Nottingham Forest and Birmingham.
Lee Carsley recently left his position at Manchester City.
Birmingham City were looking for a new manager following Gianfranco Zola's resignation, but have quickly found their man.
The two formed a strong partnership last season despite Everton struggling.
Lee Carsley made a joke about Rangers assistant manager David Weir's age.
Everton failed to win against Southampton in the Premier League on Saturday.
Kevin Mirallas has suggested that he could leave Everton in the summer transfer window.
Leeds United's youngsters beat Brentford's development squad earlier this week.
The former David Moyes pair and Lee Carsley have all had their say on Everton's current plight.