Mitchell's comments follow hot on the heels of Championship strugglers Peterborough claiming Cherries had placed an £800,000 price tag on star striker Brett Pitman.
And although Mitchell confirmed Posh director Barry Fry had made an enquiry for Pitman some time ago, the Cherries chief also said no firm offers had been tabled for any player.
Pitman, together with emerging talents Liam Feeney, Jason Pearce and Anton Robinson, could spark further interest once the transfer window reopens on Friday.
Speaking to the Daily Echo last night, Mitchell said: We've got a fantastic bunch of players and a number of them are worth a lot of money.
They are our biggest assets and, while they are young, they are appreciating assets.
I've never tried to sell an appreciating asset in my life. You try to hang on to them because they add to your future.
It would be out of the question for me to start quoting figures for our players because, as far as I'm concerned, they are appreciating assets and I would be sad to see any of them go.
But if anybody is interested in any of our players, the proper way to go about it would be to make either myself or Eddie (Howe) a firm offer and then we would have to consider it.
First and foremost, my job as chairman is to look after the finances that come into the club and to make sure there is enough to run the club.
In my eyes, it's the manager's job to make sure we perform on the pitch to get the finances to come into the club.
The only way a player would be sold or bought if we were in a position to buy, which we aren't would be through a mutual agreement between me and the manager.
It's a balancing act between satisfying supporters by getting things right on the pitch and having the finances to keep the club going. I'm looking after one part and Eddie is looking after the other so it would have to be a joint decision.
Channel Islander Pitman has thrust himself into the spotlight this season by playing a leading role in Cherries' impressive League Two campaign.
The 21-year-old netted a dramatic late penalty in Monday's 2-1 win over Torquay to register his 14th goal of the season.
Asked by the Daily Echo whether he had any comment on the latest speculation, Pitman replied politely: Like I've said before, it's out of my hands.
I can only go out, play for the team and try to score goals. Hopefully, we will keep winning football matches. Whatever else happens is out of my hands.
Pitman has been linked with several Championship and League One clubs including Peterborough, Derby County, Plymouth, Reading, Charlton Athletic, Huddersfield and Bristol Rovers.