Police were called to a building site at 20 Grosvenor Square at 3.45pm on Monday 14 April. The London ambulance service, air ambulance and fire brigade also sped to site.
A man in his early 30s, who was working on the site, was pronounced dead at the scene. Another man was taken to hospital with what are described as minor injuries.
The six-storey building, once the London headquarters of the US navy, is being converted into apartments. Reports suggest that a mini-excavator fell from the second floor.
The Health & Safety Executive has launched an investigation.
Declan Sherry, chief executive of site contractor McGee, said that the company was liaising with the victims’ families.
Jerry Swain, regional secretary for construction union Ucatt, said: “From the facts we know so far it would appear to have been a wholly avoidable accident. It is essential that there is an urgent investigation to find out how this accident occurred and lessons are learnt to ensure that a similar accident can be avoided in the future.”