Police were called to the scene near Morrisons in Dunwoody Way at 4.30pm on Wednesday, 21 June.
Another man was taken to Royal Stoke Hospital with serious injuries which are not believed to be life threatening.
It is understood Seddon Construction are building a care home on the site.
In a statement, the company said: "We're aware of an incident at our construction site on Dunwoody Way, in Crewe.
"At this early stage we are working with the emergency services and authorities in order to provide the necessary assistance.
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"We will provide further updates once more information becomes available."
Chief Superintendent Matt Welsted said Police are working alongside the Health and Safety Executive, Local Authority, and Building Control "to establish the full circumstances of the incident".
"This is a truly tragic incident, and our deepest condolences go out to the families affected at this extremely difficult time," he said.
"A residential property has been damaged as a result of the incident, the occupants were not injured and were checked over by medical professionals. They have since been relocated while the investigation continues.
"The bodies of those who have sadly lost their lives will shortly be recovered from the scene. I would ask that anybody who may have any footage of the scene, to please respect the families of those involved. Any footage that you believe could help officers with their enquiries, please send to our public contact email to allow us to effectively investigate the incident."
Police are appealing for information. Anyone who can help officers with their enquiries are asked to contact 101 quoting incident number 611 of 21 June.
(LM/MH)
