The Midlands rivals are both on the lookout for a new number one following the departures of Scott Carson from The Hawthorns to Busaspor and Villa's loss of Brad Friedel to Tottenham.
Irishman Given is set to leave Eastlands in order to seek first-team football after spending the last campaign second choice behind England number one Joe Hart.
Celtic were linked with a loan move for the 35-year-old but City are keen on a permanent deal in order to lower their wage bill.
Reports today suggest Roy Hodgson's West Brom made an offer for the former Newcastle star, but the player turned down the deal as he prefers a move to Villa Park.
The sticking point on any transfer could be the fee, with new Villa chief Alex McLeish valuing the player at £3million while City rate him closer to £5million.