Villa are poised to replace one veteran in Brad Friedel with another in 35-year-old Given who wants the chance to play first-team football again.
The two clubs have held talks and sources close to Villa are optimistic a deal will be finalised sooner rather than later.
Is that the way to Villa Park? Given is poised to leave Eastlands
Given was the No 1 at Newcastle for more than a decade before moving to Eastlands two and a half years ago.
But the emergence of England keeper Joe Hart - who played under McLeish when on loan at Birmingham - cost him his place last season.
In demand: Liverpool are interested in Downing
Personal terms are not expected to be a major stumbling block even though Given is reported to be on ?80,000 a week at City.
The expected arrival of Given, however, will not lead to the departure of Brad Guzan who had a spell on loan at Hull last season.
McLeish is keen for the American to provide back-up to Given along with Andy Marshall.
He said: 'I'd like to hold onto Brad, definitely. I don't think we will be letting him go at this particular stage.'
Villa report back for pre-season training today with McLeish set for further talks with in-demand winger Stewart Downing.
McLeish is desperate to keep the England international and has rejected offers of ?12 million and ?15 million from Liverpool.
It now remains to be seen whether Downing will force the issue by asking for a transfer.
One player who will be absent from training today is midfielder Fabian Delph who is still recovering from ankle surgery at the end of the season.
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