The 24-year-old has fallen down the pecking order under Tim Sherwood.
The report claims the defender will be allowed to leave Villa Park on loan once the deal to bring Joleon Lescott to Villa is complete.
Baker has had a torrid time with injuries in recent seasons and managed just 12 appearances last season.
Despite his fitness concerns, the former England Under-21 international was awarded a new deal earlier this summer and is still believed to be part of Tim Sherwood’s long-term plan.
The arrival of Micah Richards and the imminent addition of Lescott will push Baker to fifth choice, with Leandro Bacuna and Ciaran Clark also ahead of him in the pecking order.
Game time will be limited should he remain at Villa and a move to Bristol City would provide valuable first-team opportunities.
Burnley were also thought to hold an interest in the powerful defender – as reported by the Daily Mail – but Bristol City are now the clear favourites.