Given failed to make a single Premier League appearance last term and is eager to secure first-team football having turned 35 in April.
The former Newcastle stopper is regarded as one of the best keepers in the Premier League after his exploits with the Magpies and then City, who he joined for £5.9million in January 2009.
But Joe Hart's emergence as first-choice keeper for club and country forced Given onto the City bench for most of last season, with the 113-cap international restricted to just four cup appearances.
Although Given has two years left on his contract with City, new Villa manager Alex McLeish can offer him a regular starting role.
Villa are in the market for a new shot-stopper having seen Brad Friedel depart for Tottenham this summer.