Belgium international Fellaini, who has a release clause in his contract, has been heavily linked with a move to Arsenal, who on Thursday were backed in to sign the player with the bookmakers.
He has also repeatedly stated his desire to play in the Champions League.
But Kenwright said in Friday's edition of The Times: "I have to say he has given me every indication he doesn't want to go."
Meanwhile, there is speculation that Baines, whose contract has two years to run, could link up with former Toffees boss David Moyes at Manchester United.
But asked whether the England full-back will be offered a new deal, Kenwright said: "Absolutely right. That's started; started in that he knows we want him to stay."