But the pair rejected the settlement and Vaughan says he is powerless to wield the axe, as the financial repercussions could be dire for the club.
He told the Liverpool Daily Post: "I offered Mr Rush and Mr Aizlewood a financial settlement so they could resign from the club with dignity [which was refused] - An obvious action open to me is to dismiss them from our employment.
"I haven't carried out that course of action because of the financial implications this would bring to the club."
Chester's season has tailed off disappointingly and after winning only two of their last eleven games. And the demolition by near neaighbours Shrewsbury has prompted action by the Chester board.
Two of Town's goals in the rout were scored by former Chester transfer targets, Jamie Tolley and Ryan Lowe.
Chester now lie 21st in the League Two, ten points above the drop zone.