The manager comes off contract at the end of the season.
Kenwright revealed: "He has made a simple statement: 'Give me to the end of the season". He knows in making that statement it is not the easiest for the football club.'
"Hopefully in seven weeks' time when the last match has been played we will sit down and have the five minutes that David has alluded to for the last six months or so, that will sort out a new contract; a new adventure and will keep David Moyes at Everton football club.
"If it doesn't, the man leaves with his head held high."