The 14-man Bulls went down 31-23 to an Edinburgh outfit featuring 12 Scottish internationals in Friday night's United Rugby Championship action.
The South Africans were hampered when captain Coetzee was given his marching orders after his shoulder made contact with the head of prop Pierre Schoeman.
Speaking to rugby365.com after the match, White didn't argue against the red card, but hinted that the team may challenge it.
"Those are the laws and that is what the protocols are," said the Bulls boss.
"We just have to ensure we are not on the wrong side of the protocols. We will look at all the factors and hopefully common sense will prevail. Maybe we can argue a case.
"The best thing for us is to follow the correct channels and chat to whoever has to be chatted to.
"We will find out what their interpretations are and why it was seen like that - both ways. We are going to have to fix it.
"There is a lot of communication and documentation going out to coaches about head-on-head and that is the way it is. We have to ensure we get it right. We can't afford to play with 14 and 13 men in big games like this."
The Bulls return to Pretoria with two wins from their four-match tour up North. Next up they face an in-form Connacht, who have just one loss this season.
"I can't be disappointed with the effort, or the spirit of the group," added the World Cup-winning coach.
"We have to go home and regroup ahead of another tough game next week."