Matthews has joined existing directors Mick Power and Paul Williams in a group bidding to oust chairman Angus Johnson, following his handling of Michael Voss' sacking as coach.
The trio has been petitioning for an extraordinary general meeting (EGM) of the club's members at which it would put Johnson's position to a vote – a situation the League is desperate to avoid.
The parties met with Demetriou, AFL commission chairman Mike Fitzpatrick and deputy CEO Gillon McLachlan at AFL House on Tuesday.
No resolution was reached, and they have agreed to meet again at a later date.
Virgin, AFL extend partnership
Demetriou said the Lions had asked for a financial grant from the League, something that could be in doubt if the conflict is not resolved without an EGM.
"They've got a one-off grant before us that would help them invest in their football department," Demetriou said on Wednesday.
"Our message was very clear: we need to get this sorted so you've got a unified board that removes all the uncertainty.
"We've started that process, and we're going to re-convene later in the week."
Demetriou was in no doubt that Matthews, who coached the Lions to three straight premierships from 2001 to 2003, would be a valuable addition to the club's leadership.
"I'd love to see Leigh Matthews on the board," he said.
"He would be a great asset to any board because of his incredible experience and his football knowledge.
"One of the things the board is lacking is a football director, so he would be a welcome addition."
He said the Lions were not in as much turmoil as was being made out.
Demetriou was speaking at the announcement of a new deal with Virgin Australia, which will see the airline continue as the AFL's official carrier for the next five years.
"It's hard to believe that three years ago some were doubting Virgin's capacity to service our industry," Demetriou said.
"Our clubs have never, ever experienced the service and support that Virgin have given.
"Our players are over the moon – we get nothing but positive feedback."
Virgin CEO John Borghetti was forced to defend the airline over accusations it is over-charging Fremantle fans for flights to Melbourne this week.
"It's the old story of peak period," he said.
"We've put about six additional flights on already in addition to the normal scheduled service from Perth to Melbourne, and I think what you'll find is that the fares we've got are just as competitive as any other carrier that's flying over."
