Bild said he had admitted himself to the centre, but was asked to leave on Sunday because of aggressive behaviour towards other patients.
It was not clear whether the institution had informed the police before his release, which they are required to do in cases where a patient could pose a danger, it said.
The street where the deadly incident happened was still cordoned off as police searched the area.
At a church in the grounds of a university, Heidi Rheinsdorf said she had travelled to Leipzig from a nearby town to show support to the local community.
Wiping back tears, the 32-year-old told AFP she was “shocked” when she heard about the car-ramming, adding: “I just don’t understand why [the alleged perpetrator] did it.
“I just feel so sorry for the people.”
A 63-year-old woman and a 77-year-old man, both German citizens, were killed in the incident. Authorities have said many others were injured, without giving a precise figure.
–Agence France-Presse