Mitchell was speaking last night after Championship clubs Burnley and Preston parted company with their respective bosses, Brian Laws and Darren Ferguson.
And although Mitchell confirmed he had not received any approaches for Howe, the 33-year-old is understood to be one of the frontrunners to replace Laws.
Howe, who is six months into a three-year contract with Cherries, is still also believed to be on a short-list for the vacant manager's post at Sheffield United.
Mitchell told the Daily Echo: I'm sure they would be high on a few club's lists but they are contracted to us and I would be reluctant to see them go. I would wish them all the best in the future, but not the immediate future.
We've got the best management team in the division and one of the best squads. Having achieved so much in such a short space of time, I'm sure there are going to be clubs thinking Eddie and Jason could be the solution. But I don't think they would have any desire to leave at the moment. If they did and if the offer was too great, I would do everything to stand in their way.
Mitchell, however, is aware Cherries could be vulnerable to losing the most successful management duo in the lower leagues.
He added: People can slip in, offer the world and snap up the best ones and they are certainly in that catergory. Providing we were covered, if a Championship club came in and said they would pay them three times their wages, the decision would have to be theirs. But I have no desire whatsoever to see them go.