The 23-year-old has been attracting plenty of attention following an excellent campaign for the League One high-flyers.
The Blues and Kenny Jackett's side will have to see off competition from Huddersfield Town and Preston North End, as well as from each other, if they wish to land the 23-year-old, the Yorkshire Post adds.
Bradshaw has been integral to Walsall's emergence as a promotion contender in League One this season. He has scored nine goals in 21 appearances to help the Saddlers to third place in the division, just three points off league leaders Burton, and with a game in hand over Gillingham, who are currently denying them an automatic promotion spot.
Arguably, Birmingham's only weakness during a promising season has been the lack of goals coming from their strikers. No team in the top half has scored fewer times than Rowett's men this term, and yet they remain well in the hunt for the top six. Perhaps the acquisition of another forward may be what helps them make that next step.
While Wolves' play-off hopes are much slimmer, a great run of form could take them into the mix. Their hopes however, of moving up the table may well be hampered if they lose Benik Afobe. The former Arsenal man was last month linked with a Premier League switch (via Birmingham Mail), and should the rumours materialise into a move, Jackett may decide that a move for Bradshaw is a must if they wish to stay in contention.
At the other end of the table, both Preston and Huddersfield will be hoping that Bradshaw can strengthen their chances of remaining in the division. While it is the Terriers that are closer to the relegation zone, Simon Grayson's men have had serious problems finding the back of the net - only Bolton have scored on fewer occasions - and may need more depth up front if they wish to book a second season in the Championship.