Howe is understood to be the Eagles' number one choice to replace George Burley and is also a reported target for their south London neighbours Charlton.
The 33-year-old admitted over the weekend that he has a difficult decision to make and Mitchell is hoping he will stay at Dean Court.
"I have asked Eddie to stay, I would have been stupid not to," Mitchell told the Daily Echo.
"I have done everything I can. There are always situations in life when people will take stepping stones. If this is the one Eddie decides to take, I hope he makes a success of it but don't want him to go."
Rochdale boss Keith Hill has emerged as another potential contender for the Palace job if they miss out on Howe.