The Gresty Road side are keen to strengthen their squad with Rob Hulse, Efe Sodje, Dave Walton and Rodney Jack all departing this close season.
An initial bid of around £125,000 has already been rejected by The Hatters, and until their valuation has been met the player will not be allowed to leave.
"I don't need to sell anyone," said Palmer. "And Tonkin won't be joining Crewe unless they match our valuation of him."
Despite having their early approach rebuffed by the Edgeley Park side, manager Dario Gradi is preparing a second bid.
"The club will now make an increased offer for Tonkin as I understand he has now been transfer listed," said Gradi.
