Mulumbu is currently a free agent after being released by Norwich City at the end of last season, and it has been suggested that Birmingham boss Harry Redknapp could be keen on adding the player to his first-team squad.
According to the Birmingham Mail, Mulumbu has expressed a desire to remain in the Championship with a return to the West Midlands.
Between 2009 and 2015, the 30-year-old made 211 appearances for West Bromwich Albion in all competitions, the majority of which came in the Premier League.
Mulumbu featured on just 24 occasions during his time with Norwich, but the Congolese international remains well-respected at the second tier of English football.
The Blues have already made as many as 14 first-team signings since the end of the last campaign.