The 30-year-old looks set to leave the Premier League side this summer.
The forward is set to become a free agent in the coming weeks, and having recently enjoyed a promising loan spell with Blackburn Rovers this past season, the Mail suggests that Gary Rowett is keen on bringing the 30-year-old to St Andrew's.
However, the Mail suggests that they will have to see off competition from Blackburn, who are interested in moving for him permanently, while Rangers are also keen and newly-promoted Wigan Athletic are the favourites for his signature.
Though Graham endured a slow start to the season with his parent club, his loan spell at Ewood Park highlighted that he could certainly be a useful option for those in the Championship looking for a decent goalscorer without taking a massive chunk out of their transfer budget.
He scored seven goals in his last 16 games - including a goal in Blackburn's win over the Blues in March - so it is hardly surprising that Rowett has earmarked the striker as someone who could potentially help take some of the goalscoring burden off Clayton Donaldson's shoulders.