Motherwell boss Stuart McCall revealed last week that he expected to lose Randolph at the end of the season when his contract expires.
A number of clubs are thought to have expressed an interest in signing Randolph, but Birmingham have won the race for his signature.
Birmingham boss Lee Clark has been looking for a new goalkeeper to replace Jack Butland who is heading to Stoke next season after joining them in January before being loaned back.
Randolph has spent the last three years at Motherwell and he has impressed during his time in the SPL.
The 25-year-old is thought to have undergone a medical at St Andrew's and is set to become Birmingham's second signing of the summer following the arrival of Andrew Shinnie from Inverness on a free transfer.