Stoke, Wolves and top-flight newcomers QPR are all thought to be interested in the former Republic of Ireland international.
Former Tottenham defender Carr arrived at Birmingham from Newcastle in 2009 and has notched up 86 appearances for the Blues.
He could be the latest player to head for the exit at St Andrew's as relegated Birmingham offload members of their squad to counter their financial problems.
Roger Johnson, Craig Gardner and Barry Ferguson have already left this summer and there are other departures expected, with Scott Dann, Cameron Jerome and Nikola Zigic all attracting attention from other clubs.
Goalkeeper Ben Foster completed a season-long loan move to West Brom on Friday, while Liam Ridgewell has also been heavily linked with the Baggies.