The 24-year-old is heading to Ashton Gate to wrap up a deal with the Championship club, Press Association Sport understands.
The Robins have also been linked with Leicester right-back Liam Moore but are in the advanced stages of a deal with Baker, although it is yet to be finalised.
Bristol City are 20th in the Championship having conceded 10 goals in their opening five games since promotion last season.
Baker only signed a new four-year contract at Villa in January but has found his chances limited since Tim Sherwood replaced Paul Lambert in February.
He has made just four appearances under Sherwood, and one this season, with the imminent arrival of Joleon Lescott from West Brom set to further push him down the pecking order.
Baker has played 88 games for Villa after coming through the ranks at the club.