Adeyemi is available on a free transfer after seeing out his contract at Carrow Road earlier this summer.
The 21-year-old spent last season on loan at Brentford and he impressed during his stint at Griffin Park, making 39 appearances for the Bes as they reached the League One play-off final.
A number of clubs are thought to have expressed an interest in Adeyemi, but Birmingham are leading the race for his signature.
Blues boss Lee Clark is a big fan of Adeyemi having worked with him while he was assistant manager at Norwich.
Clark is keen to bolster his midfield options and landing Adeyemi ahead of rival clubs would be a big coup.