The Championship side are set to return with another offer.
Dundee's Greg Stewart
The Blues had made an offer of £300,000 for the 26-year-old, whose contract runs out next summer, but the Dark Blues knocked it back as they hold out for a larger fee, The Scottish Sun reports.
Birmingham manager Gary Rowett
However, The Scottish Sun claims that Gary Rowett is unlikely to be put off by the setback as he plans to make an improved offer, and with the report stating that Dundee may be willing to accept around £375,000 with add-ons, it seems very possible that their next bid could be much more appealing to Stewart's current side.
There is no doubt that Birmingham are in need of adding some fire power this summer as they look to build on the promise they showed last season.
Clayton Donaldson was Birmingham's top goalscorer last season
While Clayton Donaldson proved last term that he is capable of scoring plenty of goals - he finished the campaign with 11 league strikes - he was the only man in the club's squad to reach double figures, while their second highest scorer, Jon Toral, will not be playing at St Andrew's this year after Arsenal loaned him out to Granada.
And with their other recognised strikers failing to contribute a league goal since they joined the Midlands side, Rowett will surely feel that signing Stewart - who has 22 league goals across the last two seasons - is something he needs to do, especially as his first bid is seemingly not that far off Dundee's valuation.