Violence amongst travelling supporters during the second half was documented by Hearts officials, who are now debating whether to reduce the number of tickets issued to Celtic next season from 3,500 to 2,100.
Followers of the Glasgow club were seen assaulting police and stewards in the Roseburn Stand, whilst Hearts' ballboys had to be withdrawn from the trackside in front of the away end after being struck by missiles.
Hearts are awaiting news of the SPL's investigation into how a supporter managed to enter the technical area to attack the Celtic manager Neil Lennon during the second half.
They could be issued with a fine as a result, however, they are determined to cut the potential for trouble when Celtic next visit Edinburgh.
Slashing the number of away tickets is one option being discussed although Celtic officials must be consulted before any such move can be taken. In 2006, Hearts cut away allocations for both Old Firm clubs in the Roseburn Stand in order to satisfy demand from their own fans.
The club is considering a similar move this time with a Tynecastle spokesperson stating a final decision won't be taken until the SPL's investigation is completed.
"Nothing has been decided yet and a final decision will only be made following the results of the inquiry into the events of last Wednesday night, which is currently on-going," said the spokesperson.