Hart joined the Blues on a season-long loan from Manchester City in the summer and recent performances have led to talk of the 22-year-old jumping to the front of the queue to be England's number one at the World Cup.
While that may be premature, Hart has certainly done his chances no harm having been instrumental in the club's current 11-game unbeaten run in the Barclays Premier League.
The former Shrewsbury youngster produced another fine display as Birmingham drew 0-0 at Nottingham Forest in the FA Cup third round on Saturday, his fingertip save to thwart Robert Earnshaw particularly impressive.
City boy: Hart
McLeish admits he would like to keep Hart at St Andrew's permanently but the Scot is resigned to the fact that he will be returning to Manchester City at the end of the season.
'We've asked them and there is no chance,' he said. 'Man City have said he's not for sale.
'To be fair that has always been the way as they said that in the summer. That has always been their stance. He made a couple of good saves again and played his part.'
McLeish added: 'When he first arrived there were people telling me after one or two games I should maybe be thinking about making a change. They obviously couldn't see the good things he was doing and look at him now.
'We discussed it and changed a couple of things and he's improved. He's a guy who wants to improve, wants to learn and wants to be the best and it always helps when you have got a student who is receptive.'
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