The match was called off as spectators were still entering the ground
Yorkshire will find out on Thursday whether they have been thrown out of the Twenty20 Cup for fielding an ineligible player.
An England and Wales Cricket Board discipline commission will meet at Old Trafford to discuss the matter.
Azeem Rafiq, 17, played in a group match but was not registered and did not have a British passport.
The quarter-final between Durham and Yorkshire was postponed just before the start when the matter came to light.
Durham chief executive David Harker believes his county should advance to the Twenty20 finals day as the only "fair" outcome to their quarter-final fiasco.
"What must not happen is that we are further damaged by this process," Harker told BBC Radio 5 Live.
"The only option that enables that is for us to proceed to the finals day."
Harker said he realised other teams may feel entitled to a second chance at rejoining the competition but he felt going straight through to the finals day was the only way to avoid his team being punished for the actions of others.
"I accept it's not entirely satisfactory but any other option just causes us more problems. Fixture congestion is a real problem.
Interview: Durham's David Harker
"We have recruited players especially for Twenty20, two of whom wouldn't be available for the rearranged fixture, so we would be fielding a weakened side on the back of a congested fixture list through no fault of our own.
"It isn't our desire to throw Yorkshire out but I can only see it as being the only practical and fair solution".
Yorkshire chief executive Stewart Regan has already confirmed the county failed to register the teenage spinner correctly - and that he took full responsibility for the debacle.
"Ultimately, the buck stops with me," Regan told BBC Radio 5 Live. "I am responsible for running the club."
Asked whether he would consider resigning over the matter, Regan replied: "I don't think making a scapegoat of someone is what this matter is all about."
Seven thousand fans were gathered at the Riverside on Monday for the scheduled 1740 BST start when news of the postponement filtered through barely half an hour before play was set to start.
Regan claims the ECB was willing to backdate Rafiq's registration but then discovered he did not hold a British passport, leading to the match being called off.
The lack of a passport has not stopped him being picked for the England Under-15 and Under-16 sides but the ECB said rules governing schoolboy cricket do not require players to be born in the country they represent.
However, Rafiq was only eligible to play for Yorkshire at levels below the first team.
Rafiq made his Yorkshire debut in their final group match against Nottinghamshire, which the panel may order to be replayed.
If Yorkshire are thrown out, Notts may feel they should replace them in the quarter-final. However, they have said they will not make any public comment until they hear the ECB's decision.
Glamorgan finished third in their regional league but did not go through as the other two third-placed teams, Yorkshire and Kent, had more points.
And chairman Paul Russell said his team would leap at the chance to re-enter the tournament.
"There must be a slim chance that we will get back into the tournament," he said.
"If Yorkshire are thrown out then Glamorgan would seize the chance with all the energy we could muster."
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