The 24-year-old, who has had a stuttered career due to off-field issues, has been training with the Championship outfit for the last three weeks.
Morrison remains on Lazio's books following his 2015 switch from West Ham United, but he has not featured for the Italian club since April last year.
Birmingham manager Harry Redknapp is familiar with the midfielder's qualities having coached him at Queens Park Rangers in 2014, but his attempts to sign him are not going to plan.
According to Sky Sports News, the Blues' request for a loan move has been rebuffed by Lazio, despite the former Manchester United academy player featuring in two of Birmingham's pre-season friendlies this summer.
After leaving the Red Devils in 2012, Morrison has signed for West Ham United and Lazio, and also had loan stints at Birmingham, QPR and Cardiff City.