The Blues sacked Steve Cotterill on Saturday following a 2-1 defeat to Nottingham Forest - the club's fifth consecutive league loss and a result which leaves Birmingham two points from safety in the Championship table.
Monk immediately emerged as the frontrunner to succeed Cotterill, and Sky Sports News reports that discussions have moved quickly.
Confirmation of Monk's appointment is expected within the next 24 hours, with the Birmingham hierarchy eager to get the deal done in time for Monk to take training on Monday.
Should that be the case then Monk's first match as Birmingham manager would come against Middlesbrough - the club that sacked him in December after only six months at the helm.
Monk has also spent time in charge of Swansea City and Leeds United during his managerial career.