Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
Redmond, 19, has enjoyed an impressive end to this year's campaign as the Blues steered themselves well clear of relegation trouble with six wins in 11 games.
The youngster has just one year left on his contract and is attracting attention from a number of Premier League clubs, but Clark insists only a hugely substantial bid will prevent Redmond spending another season at St Andrew's.
Clark told the Birmingham Mail: "It would be a crazy decision to cash in on him.
"It would have to take big money, a fantastic offer, for us to part with him.
"Can we keep him? Yes, of course. Why not? He's a superb player. He's a local lad and he's enjoying his football."