The Blues forward heads to Shropshire in search of First Team football having found opportunities limited since his arrival in the winter transfer window.
Cosgrove, 24, signed from Aberdeen on 31 January 2021 and to date has recorded 12 appearances for Blues, ten of them as a substitute.
He will link up with the former Birmingham City boss, Steve Cotterill, for the remainder of the season.
The Shrews will be hoping Cosgrove can enhance their attacking options having fired a blank in their opening day defeat to Burton Albion.
Good luck, Sam!