The former Hearts skipper was offered a deal earlier this week and has returned to his native Lithuania to consider the offer.
Zaliukas - who left the Jambos in the summer - impressed the manager while training with the Light Blues and McCoist should know in the next 48 hours or so whether or not he has landed the defender.
He said: "Marius has gone back home. We have put an offer to him and we would be very hopeful of speaking to him in the next couple of days to see whether or not he will take that offer up."
If the answer is no, McCoist says he has other options to bolster his defence, with talks still ongoing with Bosnia international Boris Pandza.
He added: "I know we are still in negotiations and speaking to Boris Pandza's agent.
"I know Craig (Mather, chief executive) has been speaking to him.
"I will speak to Craig later today to see if there are any further developments. We are still pursuing a couple of lines in central defence."
Asked if there was a chance of signing both players, McCoist said: "I don't know. I couldn't say it would be both or one or the other.
"I would have to wait and see what progress has been made in terms of the details of the offers."
Greek defender Anestis Argyriou is the latest player to head for the Ibrox exit door and McCoist says more could follow.
He said: "There possibly could be. I think that's another four boys who have left the club in the last month or so.
"I couldn't give you specific names but, like most clubs, there will be activity and players hopefully coming in as well as leaving."
Meanwhile, McCoist declined to be drawn on where the Ramsdens Cup final should be held this year if semi-finalists Rangers make the last two.
Competition organisers would likely seek a bigger than usual venue, given the size of the Light Blues support.
With Hampden unavailable due to the Commonwealth Games, Celtic Park and even Ibrox itself could be possible alternatives.
McCoist said: "I would be far happier having this conversation if we are in the final. I'll probably save it for then.
"At this moment in time, it's for the powers that be to discuss where the final will be.
"I'll certainly give you my tuppence-worth if we are fortunate enough to beat Stenhousemuir away.
"To be honest, if you were asking most of the players, they would want to play in a bigger stadium with a full house.
"In terms of the sponsors, I would imagine they would want to make it as big an occasion as possible. These are factors that would have to come into it as well."
- Sports & Recreation