An inquest into the death of carpenter Vincent Dooley recorded a verdict of misadventure. He had been working at McAlpine's 280 Bishopgate site in the City of London.
The decision comes just a week after the start of a TUC crackdown on safety in construction and after seven site workers died in seven days last month.
The City of London Coroner's Court heard last Wednesday how Dooley fell 3.5 m from the 14th floor of the building through to the 13th floor while working for electrical engineer McDermott Brothers.
An HSE source said an investigation into the circumstances surrounding Dooley's death was continuing.
He said: "The investigation will be concluded within the next two months and then a decision as to whether Sir Robert McAlpine will be prosecuted will be made."
Coroner Douglas Chambers explained how Vincent Dooley had suffered injuries to the head and elsewhere after the fall. He asked the jury to reconvene after they twice returned split verdicts on the cause of death.
The inquest heard that safety harnesses were available in the site office, but that Vincent Dooley was not wearing one. There was confusion as to which of two uncovered holes Dooley fell through on the site.
Dooley's death was one of five in nine days during December 2000. Another of the deaths has resulted in a prosecution. Yorkshire-based contractor Billington Structures was fined £200,000 after the death of employee Andrew Kitchen in December.
Kitchen, 43, from Newark, had been working on a subcontract for Rowland Construction.
He was removing steelwork to make way for a corner stand at Sheffield United's football ground when a wheel of the 15-tonne elevated platform he was working on went through a manhole cover and caused the structure to topple over.