North West Ambulance Service said three of its crews and an air ambulance’s critical care team treated four people at the scene.
A Greater Manchester Fire and Rescue Service spokeswoman said firefighters had used specialist cutting equipment to free one of the injured people from the bus.
The incident led to major disruption on Friday afternoon and evening.
Piccadilly Gardens, Market Street and Shudehill tram stops were closed while the bus station was also shut.
Mr Karney, who has been a Manchester City Councillor since 1979, said: “The whole city centre was brought to a standstill so that’s why I’m calling for an inquiry into what went on.”
He said any inquiry would be overseen by transport authorities, with involvement from the council.
Tram operators KeolisAmey Metrolink and bus operator Stagecoach have also been assisting investigators.
Incidents involving trams are investigated by the independent Rail Accident Investigation Branch., external
The has approached the RAIB for comment.
