Chester reveal Toffee interest
The 22-year-old rejected the offer of a new one-year contract with The Toffees last month and the former England Under-21 international has now attracted the interest of the League Two new boys.
"We have contacted Everton about Peter's availability, and if we receive a positive reaction, we will take the matter further," said Vaughan.
Despite being out of contract, Clarke would not be able to move on from Goodison Park for free, as he is under 24 years of age.
The news comes as Vaughan revealed that the club are in talks with an unnamed group of businessmen from Dubai over possible investment in the Deva Stadium outfit.
Meanwhile, Chester boss Mark Wright ran the rule over four trialists in Tuesday's pre-season draw at Heswall.
Argentine defender Joseph Belfortti, former Southampton man in Australian Jay Lucas, French striker Gregory Thil and midfielder Stuart Roberts were all given the chance to show their abilities to Wright.