Posh turned down those Bournemouth bids, believed to be of £1 million and £1.5 million for a player out of contract in June.
Fry said: “Birmingham got much closer with their offer to what we want, but when I told Siriki about it he turned his nose up.
“He probably prefers Bournemouth, but there’s no way he’s going there for what they’ve offered.
“The owners have carried our club financially for two years with hardly any income and yet they, like me and the manager, would rather Siriki stayed and helped us stay up.”
Dembele has been on the transfer list for over 12 months. He can leave for nothing at the end of this season.
Posh are understood to want a package around the £4-5million mark for Dembele which includes bonuses based on future achievements.