Mawhinney wants administrators KPMG to restore a Company Voluntary Agreement - a deal with Leeds' creditors - before any discussions take place about giving them permission to compete in League One.
He said: "We have to resolve the situation and we have to resolve it soon because the season starts in eight days' time. On Tuesday, we had a four-hour meeting with Leeds and the administrators. We all agreed that the best way forward would be to see if the administrators could reconstitute the CVA.
"Part of our policy is to do whatever we can, within the rules, to ensure the continuation of our member clubs through administration and that remains true of Leeds United.
"I don't have a crystal ball to see into the future but I am an optimist. If the CVA is not granted, we will cross that bridge when we get to it."