The 35-year-old, who led the Cherries to promotion this season having returned from Turf Moor for a second spell, has been strongly linked with the vacant managerial role at Goodison Park.
And although Mitchell is not surprised by speculation linking Howe with a move to the top flight he is determined to retain his services.
Mitchell told the Daily Echo: "I think he is worthy of a Premier League job. We don't want to lose him but he is a fantastic manager and you can't take that away from him. His record speaks for itself so he is bound to be linked with Premier League clubs.
"If I was a boss of a Premier League club, I would be looking for somebody young and dynamic and someone who could bed in and stay for a long period of time. I am hardly surprised that, because of his record, he is linked with Premier League jobs.
"It is paramount I do everything I can to keep him and I will do so. All I can do is my best. We speak regularly, he loves what he is doing and we love him doing it."