The Gers could be without no fewer than FIVE first-team regulars for the match at Easter Road on Sunday afternoon.
Ian Black, Jon Daly, Nicky Law, Andy Little and David Templeton are all struggling to make the meeting with the Kirkcaldy club.
McCoist admitted he would give every member of the influential quintet as much time as possible to prove their fitness.
But he revealed: "It is not great at the moment. We're giving everyone right up to the last minute.
"The biggest doubts are Templeton and Little. Black, Law and Daly have a better chance.
"But we'll give everyone right up until Friday - and maybe even Saturday as well."
Rangers will play the William Hill Scottish Cup semi-final against Dundee United at Ibrox six days after the Ramsdens Cup final.
But McCoist stressed he would pick the strongest side he has available to him and will not worry about the United match.
He said: "We want to win the cup final. We've got to win on Sunday.
"Yes, the game against United the following Saturday is a massive game for us, but I don't think we are in the position to start looking and picking and choosing games.
"We're just not well-off enough in numbers to be able to do that. Plus I think the importance of the game on Sunday determines that we have to choose our strongest team come what may.
"There is no chance of the players' minds wandering to the Dundee United match. None. They can do that on Sunday night or Monday morning."
Meanwhile, McCoist has expressed optimism Lewis Macleod will make a full recovery from the viral condition that has sidelined him this season.
The Gers gaffer revealed at the weekend that the complaint had affected the muscles surrounding his heart.
He said: "The results of his test have been sent down south. We are waiting on feedback from the medical people down there. Fingers crossed.
"You feel sorry for him because he feels fine. It's a strange one. A doc explained it to me as him having an uncommon reaction to a common virus.
"It's affected some of the muscles in his body and some surrounding his heart. The thing has got better so we have to remain positive and upbeat. You feel for the wee man. But he's doing bits and pieces."