Howe, replaced Jimmy Quinn as Bournemouth manager on New Year's Day in 2008, and led the Cherries to promotion from League Two last season.
"Eddie's always up for the challenge and believes in himself and that's what's propelled him to where he is now," Mitchell added.
"Money has to be a consideration as money gives you security, he's got to digest everybody's opinions and his own thoughts and make a difficult decision."
Bournemouth are currently second in League One, just four points behind leaders Brighton.
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