Ronald Koeman lost his job as the Everton boss last week after a poor start left the club in the relegation zone after nine Premier League games.
David Unsworth has been named as the interim boss but has admitted he would like to take the job on a permanent basis.
Sean Dyche has been heavily linked with the role after a strong start to the campaign with Burnley.
He has worked wonders with a limited budget in his five years in charge of the Clarets and has established the club as a credible Premier League team.
But Carragher has thrown a spanner in the works by admitting on social media he would like to take the job at Everton.
In response to a fan on Twitter who asked whether he would apply, Carragher toyed with the idea.
He posted: “Who says I haven’t? I’d love the job!!”
Speaking on Sky Sports’ Monday Night Football programme, the Liverpool legend said Dyche can’t turn down Everton if they do come knocking.
“If he has been, or gets offered the Everton job, he has to take it. He has to,” he said.
“Maybe Burnley fans don’t want to hear that, but he has to.
“Sean Dyche won’t go to a team in the Champions League in this country. Everton are the next step.”