Hart, who spent a season on loan at St Andrew's earlier in his career, is now 2/1 to head to Birmingham this summer, with any MLS club (4/1), Nice and Rangers (both 8/1) also listed among the likely destinations.
The 31-year-old is expected to leave the Premier League champions this summer, after entering the final year of his contract - the past two of which have been spent on loan to other clubs.
And Birmingham could be looking for reinforcements in goal, with David Stockdale and Tomasz Kuszczak thought to be unwanted at St Andrew's.
That said, the Mirror reported earlier this week that the Blues have been placed under a transfer embargo, which if true would make Hart's signing unlikely.
And even if that proves to be false, the former England number one's Manchester City wages - which are around £87,000 a week according to the MEN - could prove difficult for Birmingham to afford.
Could Hart leave Manchester City for Birmingham?