The 34-year-old sees his current contract expire at the end of the campaign with reports hinting he had turned down an offer from the club.
But McCoist met the shot stopper yesterday prior to training and was encouraged by what he heard.
It is now hoped that Alexander's agent can speak with Rangers chief executive Charles Green and head of football administration Andrew Dickson in order to bring talks to a close.
Speaking about the former Scotland internationalist, Light Blues boss McCoist said: "I spoke to Neil yesterday morning. We have had good discussions and I am obviously very hopeful him and his people can speak to Andrew and Charles.
"Neil has shown great loyalty to the club and he has never, ever been a problem. He's a great trainer and his professionalism is fantastic.
"He works really well with Jim Stewart (Rangers goalkeeping coach) so it would be crazy not to encourage someone like that to stay."
McCoist also hinted Fran Sandaza could start against Berwick Rangers this weekend with Lee McCulloch struggling to be fit for the game because of an ankle problem.
The Rangers skipper admitted he might have to miss matches if the injury which has been hampering him for a couple of weeks doesn't improve.
And having scored in yesterday's 2-1 closed-doors defeat to Carlisle United, Sandaza could be in line to step in to the first team.
Having recovered from a broken cheekbone, Sandaza has played in three games but has yet to add to the one competitive goal he has scored this term against East Stirlingshire.
McCoist told the official Rangers website: "I thought Fran played reasonably well down in Annan and he could and probably should have scored.
"There's perhaps an opportunity to play him through the middle while Andy Little and David Templeton can do that too.
"We've got one or two options there and in Fran's case, all he is lacking at the moment is a goal.
"He runs the channels very well, holds the ball up and brings people in so a goal would do a lot for him."