Bradford's Andrew Davies, Luke Oliver and goalkeeper Jon McLaughlin were red-carded, along with Crawley's Pablo Mills and substitute Claude Davis.
Referee Iain Williamson called the players into his dressing room when punches were thrown after the game.
The FA will review video evidence of Tuesday's match and talk to both clubs.
Crawley manager Steve Evans said: "If people have acted out of turn they need to be dealt with."
Bradford manager Phil Parkinson added: "I have never been in a situation before where a ref has come into the dressing room, pulled players on one side and sent them off."
A spokesman for the club also said they were in the process of investigating the incident "and gathering all the facts from the players".
"We have received a disciplinary notice from the Football Association.
"We are taking the incident very seriously and if the chairman or manager believes the incident warrants fines to the players, we will issue them."
A statement from Crawley read: "Regarding the unfortunate events which took place at yesterday's game at Bradford City after the final whistle was blown, Crawley Town cannot make any further comment until the club has had the opportunity to view the video footage in detail."
The FA rulebook states that players dismissed for violent conduct receive an automatic three-match ban.
Bradford's Davies should now be suspended for five games as he had twice been dismissed earlier in the season, with each red card incurring a further match ban.
Due to the scale of the confrontation, additional bans could be handed out to the players, and the clubs could be fined for failing to control their players.
In 2002, Sheffield United were fined £10,000 by the governing body after three players were sent off following a series of fights and illegal tackles in a match with West Bromwich Albion which was abandoned after 82 minutes when the Blades were reduced to six men due to dismissals and injury.
Bradford and Crawley could both also issue internal fines and suspensions following their own independent investigations.