Brown featured in an under-21 game for the Hammers last night against Brighton and Hove Albion, a 2-0 victory.
The midfielder is an England under-20 international who has made one appearance for the Blues this season.
Speaking to Birmingham City's official website, Rowett explained the decision to let Brown go on trial.
"Because Reece has not been involved in our first team, both parties just felt this was a good opportunity for him to go down there to train and play."
The 19-year-old is not eligible to play in first team games for West Ham, and the trial will continue over the rest of the month, ending on January 27.
Brown's current contract with Birmingham City expires in June, and if West Ham are keen on signing him then they are able to put forward a pre-contract agreement to sign him on a Bosman, or pay the Championship side a fee to sign him now, to avoid a tribunal.