Brymor Construction is the main contractor on the Clarion Housing Group job.
The 30-year-old male victim was rushed to hospital but later pronounced dead.
He has been named locally as Scott Grimes and a fund raising page has been set up by his friends here.
A spokesperson for Hampshire Constabulary told the Basingstoke Gazette: “We were called at 9.46am today (February 15) after a 30-year-old man was found injured on a construction site in Wiltshire Crescent, Basingstoke.
“He was taken to hospital, where he was sadly pronounced dead. His family have been informed and are being supported.
“Officers are in attendance at the site and enquiries are ongoing as to the exact circumstances of his death, in liaison with Health and Safety Executive.”
A spokesman for Brymor Construction said: “Regrettably an incident occurred yesterday, at our Wiltshire Crescent site, where an operative was found unresponsive.
“Sadly despite first aid being administered by our team and the emergency health professionals, they were unable to resuscitate him.
“The matter is being investigated by the relevant authorities and the company continues to assist them with their inquiries.
“The thoughts of everyone at Brymor remain with his family and friends at this time.”