Dawson yesterday quit the Bucks in order to join their National League North rivals, having informed boss Gavin Cowan over the weekend of his desire to find a club closer to his Manchester home.
Telford claim Chester then made contact and held talks directly with Dawson on Monday, without making an official seven-day approach.
That has prompted the Bucks to report their rivals to the FA and the National League.
A statement said: “The club received information to suggest Chester had already made contact with the player regarding a move and that Adam was meeting them to agree terms without club receiving an official seven day approach to allow permission.”
Dawson, who joined the Bucks in the summer, scored one goal in 14 appearances for the club.
Chester have denied any wrongdoing in a statement issued on their website, insisting that the club have acted within guidelines set out by the FA involving transfers:
"Chester Football Club are aware of a statement made today by AFC Telford United regarding Adam Dawson."
"As a club we strenuously deny any wrongdoing and believe we have acted within all FA guidelines throughout the transfer negotiations with both Adam and his previous Club. We await FA ratification of this deal."