In the absence of the Irishman, Brad Guzan has impressed the manager to the extent the Scot has told the 35-year-old he is no longer the club's definitive No.1 stopper
Whilst the Scotsman is delighted that Given has returned to training after his stint on the sidelines, he has stated that the impressive form of back-up stopper Guzan have been too good to ignore.
McLeish told the club’s official website: "Shay is not far away. He's champing on the door to be involved this weekend against Everton.
"Whether he is going to make it, we will have to wait and see. I can't give you that news at the moment.
"He has made great progress. He is training with the group and it's a case of how sharp he's feeling and how confident he's feeling before we make a decision about him coming back into the team.
"There are no guarantees of that obviously because of the way Brad has played.”
The manager also singled out the American's performance in the FA Cup win over Bristol Rovers as an example of his quality, even though he was denied a clean sheet by a last minute consolation in the 3-1 victory.
"Brad has been fantastic. I felt great for him when he made that penalty save against Bristol Rovers in the last game although I was gutted for him that he didn't get the shutout.
"I know all goalkeepers crave clean sheets. It is always a gold star for the goalkeeper when there's a shutout but unfortunately we let a goal in.
"But what a fantastic save for the penalty. The experts we have here - like coach Terry Gennoe - tell me that Brad has a great record when it comes to penalties.
"He's got a good knack at that, going back to the Peace Cup tournament in Spain a few pre-seasons ago. Well done Brad.
"It was good to see him saving that penalty although, in all honesty, he didn't deserve to let any goals in that game."
Given has been on the sidelines since he picked up a hamstring injury in Villa's game against Manchester United early in December, handing Guzan seven games to prove his worth.