The 67-year-old has been told by the Scot that he wants to discuss his future at the end of the season, but agrees that should he leave he will do so with his head held high
The Scot has been linked with managerial positions in the top four, which has made the 67-year-old more determined to keep him at the club.
Kenwright told Sky Sports News: "He's made a simple statement, give me to the end of the season.
"He knows in making that statement it's not the easiest one for the football club and he's acknowledged that to me. But he also knows the respect I have for him.
"So hopefully in seven weeks when the last match is played we can 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 keep David at the football club.
"If it doesn't, the man leaves with his head held high."
Should he leave, Kenwright would have to start looking for a replacement manager, the first time he has had to do so in a decade, but he is not worried about that just yet.
He added: "Does the phone ring? Yes. What is my response? My response is very simple - Everton Football Club has its manager."
New ownership could also be on the horizon, after the chairman admitted he is willing to listen to offers for the club.
He added: "They (the fans) don't want another Bill Kenwright. They want someone who can take it forward.
"There will be someone out there. This week, yeah, some more have come forward. You've got to remember, everyone knows the football club is for sale."