It is also commencing work on new projects to build emergency COVID-19 wards.
The announcement was made following extensive checks and agreements with its project teams, clients and supply chain.
Construction activity at BAM Construct sites across the UK was paused last week.
The industry and government have jointly defined new ways for sites to operate, including avoiding public transport and maintaining the two-metre social distancing guidance.
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In a statement, BAM said: “Our teams around the country have been reviewing and redesigning work at all sites.
“This has allowed discussions to be had with our clients, sub-contractors and those employees who are willing and able to work at sites, about re-opening.
“To support this, we have significantly reduced the number of people who will be present on our sites to help us work safely.”
Working arrangements were said to be under constant review, with the closed sites mainly in London, Scotland and the Midlands.
“…We continue to work on how these can be safely re-opened at a future date.”
BAM has also been working closely with the NHS, Department for Health and the UK government on numerous critical healthcare schemes around the country.
This includes building new or adapting current facilities to treat COVID-19 patients.
“Buildings are central to people’s lives, and at this time they are also central to saving people’s lives,” stated James Wimpenny, chief executive at BAM Construct UK.
“These are the most extraordinary and concerning times for us all.
“We are all in this together, working hard to save lives and livelihoods.
“I want to thank everyone at BAM for everything they are doing to keep themselves and their families safe, and for responding so well to rapidly changing circumstances.”