Merseyside Police confirmed that a 53-year-old man died at the scene and a second man, 18, was taken to hospital where he remains in a critical condition.
The victims have been named locally as David Bottomley and his son Clayton, both from Castleford in West Yorkshire.
It is understood that a mast climber failed dropping the victims several floors at the Unity Street site where Laing O’Rourke is main contractor.
A Laing O’Rourke spokesperson told the BBC: “We can confirm there was an incident at the Unity Building yesterday afternoon in which two men were seriously injured.
“Our Health and Safety team is at the site and we and our sub-contractors will cooperate fully with the investigation into what happened.”