Five players were sent off for violent conduct after the final whistle - Crawley's Pablo Mills and Claude Davis, along with Bradford's Jon McLaughlin, Luke Oliver and Andrew Davies.
The FA will contact both clubs once they have studied referee Iain Williamson's report. It's understood that charges, should any arise, could be levelled before the end of the week.
Both managers admitted after the game, which Crawley won 2-1, that they could take action to punish their own players.
Bradford boss Phil Parkinson said: "Both clubs have paid a heavy price.
"I can't believe this has happened. 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. He wouldn't allow me in there!
"I will look at the DVD and if any of my players are guilty of throwing punches they will pay heavy fines."
Crawley manager Steve Evans claimed: "I only saw the Bradford player Andrew Davies run towards my players and throw a punch. I think it was handbags, with a lot of pushing and shoving, but it was disappointing to see.
"If people have acted out of turn they need to be dealt with.
"It is a shame because it takes away from what was an outstanding performance."