The three main Gers fans' groups, the Assembly, the Association and the Trust, have warned they may "disengage with the club".
And in a statement that backed the requisitioners' bid to be elected to the board, they stressed that would have "disastrous" financial consequences.
The revelation comes at the end of a week when protest group The Sons of Struth emailed all the major shareholders in Rangers.
SoS, who will stage a demonstration at the SPFL League One match with Ayr at Ibrox today, threatened to stop buying season tickets and official club merchandise. They have vowed to take drastic action if the current board survives what promnises to be an explosive AGM on December 19.
McCoist said: "I will never criticise anybody's right to object or protest or say what they believe in. That is one of the things that this country has always prided itself on.
"But when it starts affecting the support of the team it would disappoint me a little bit.
"All I ask, and all the supporters have given us, is incredible support for the team. I honestly think that would continue no matter what grievances they have with the club or whoever is running the club.
"I can't speak for the supporters, but I would think if you asked any of them, their support of the team would always, always be there.
"They might stage one or two protests to make points. As I say, I'm not going to criticise that, far from it. They're free to protest in any ways, shape or form they like as long as it's legal.
"But all I would ask them to continue to do is support the team and they have been fantastic up until now.
"I would obviously say to them that the club and the team need the supporters as much or even more than they ever have."
McCoist is confident that his players will continue to perform well as off-field unrest escalates in the build-up to the AGM.
He said: "The players totally understand the job they have got to do and that is win games and provide entertainment for the supporters.
"That is their job. They need to continue to do that and let the supporters, the board, the directors all do what they have got to do. Then we will see what happens after the AGM."