The 31-year-old was let go by the Championship side at the start of the summer.
The 31-year-old was released by the Villains following their relegation to the Championship, and it appears that Gary Rowett is considering whether to offer him the opportunity to stay in the Midlands.
Birmingham manager Gary Rowett
The Mail reports that the Blues have taken former Swansea full-back Dwight Tiendalli with them on their pre-season tour of Marbella on a trial basis, but Rowett is keeping his options open and is keeping Richardson on his radar.
Should Birmingham pursue a move for the former Manchester United trainee, the decision will surely be met with a mixed response by the club's supporters given his spell at Villa Park.
The defender spent two years with the Villains and was one of a number of experienced players who failed to find their best form at any stage last season as the team went down largely without a fight.
However, he would bring a versatility that would make him useful to Birmingham's squad as he is capable of playing at the back or on the wing, and his arrival on a free transfer means that there is little risk in bringing him to St Andrew's on a short-team deal.
And with many fans likely to be disgruntled about the switch - especially, as he has also played for West Brom - he may arrive motivated by wanting to prove a point by helping the side mount a promotion challenge.