The 21-year-old defender is out of contract at Selhurst Park but Palace would receive a compensation fee due to his age.
Manchester United, Newcastle, Reading and West Ham have all been linked with a move for the England Under-21 international but a fee for his services remains a sticking point.
Parish said: "We need to do a deal with Clyne. There is a bit of action in the background and we know there is a lot of interest.
"It is just about sorting out who it is best to go to and who will put the right money on the table. We're all working to get an outcome.
"Clubs don't want to go down the tribunal route. It is a game of cat and mouse on the fee - we've just got to let that play out."