Housing Secretary Robert Jenrick told Sky News that construction on open air sites can continue but workers should follow social distancing advice.
Some construction companies are choosing to close anyway.
Listed housebuilders Persimmon and Bellway confirmed on Wednesday morning they were beginning a phased shutdown of their sites, despite the regulations now allowing them to operate.
Their updates followed the lead of rivals Taylor Wimpey and Barratt Developments who had already announced a suspension of ordinary activity.
A Cabinet Office spokesman told Sky News those who can not work from home should go to work if they are feeling well.
He told Kay Burley@Breakfast: "If you don't follow those rules there will be enforcement action taken against you".
The health minister urged those "well and able to do so safely", to sign up for the new scheme.
As part of new measures announced on Monday, Hancock told workers to only leave their houses to go out to work "where this is absolutely necessary and can not be done from home". He said those who can not work from home should go to work "to keep the country running".
He said construction workers were among those who could keep working - as long as they could remain two metres apart at all times.
There have been over 8,000 confirmed cases of coronavirus in the United Kingdom, and 422 deaths, according to the latest Department of Health figures.
But many construction workers have reported commuting to busy building sites on packed London Underground trains leading to calls from workers, politicians, and unions for the government to prioritise public health over the economy with many saying the work is not essential and is putting people at risk.
Hancock said Tuesday that as many as 11,788 had responded to say they will return to the NHS to assist, including 2,660 doctors, 6,147 nurses, and 2,500 pharmacists and other staff. He added that some 18,700 final-year student nurses and 5,500 final-year medical students will "move to the frontline" next week to help combat the CCP virus.
Mr Hancock said: "I pay tribute to each and every one of those who is returning to the NHS at its hour of need".