The England Under 21 defender put pen-to-paper on a four-year deal at St Mary's last week, which he can do as he was out of contract.
The two clubs are now in talks as Clyne cannot move for free as he is only 21, but Championship Palace are not happy with what Premiership Southampton are offering.
"I don't know what the "reported" compensation offer was for Clyney but I know what the real offer is and yes, Southampton are taking the mick," he told cpfc.org.
"We think that we have developed one of Britain's finest young defenders through our academy and then given him great experience playing first-team football.
"He's certainly got the ability to be a very good Premier League player and we will not be accepting their offer but going to tribunal.
"It's a no-brainer for us because what they have offered is so low.
"We got £750,000 for Bostock who had played one league game whereas Nathaniel's development is way in advance of that and so should be the fee.
"It's annoying that it can't be sorted out sensibly. Nevertheless the compensation should be far more than they have offered."