Roberto Martinez, Malky Mackay, Gus Poyet and Neil Lennon have all been linked to the role at Goodison Park.
However, there is a groundswell of opinion that Everton should look closer to home and reports are claiming that Kenwright is already looking at internal candidates.
Phil Neville, David Weir, Steve Round, Duncan Ferguson and Alan Stubbs are all said to be potential candidates to help fill the void left by Moyes.
"I have got people in mind," Kenwright is quoted as saying by The People.
"A lot of those mentioned are not too far away.
"Everton have got a type, it's a very hard working industrial football club."