The union wants workers covered regardless of their employment status or where their employer sits in the supply chain.
Industry benefits scheme B&CE provides accident and life cover paid for by the employer which costs just £1.49 a week.
If a member dies at work their family receives £80,000 and if they die while not at work the benefit is £40,000.
Unite national officer for construction Jerry Swain, said: “While no amount of money can ever compensate for losing a loved one, if workers are part of the B&CE scheme, it at least means that the family grieving the loss of a loved one, will have one less thing to worry about.
“It is essential that clients and major contractors take action to ensure that all workers are covered by the B&CE scheme regardless of who engages them or the employment status they are given.
“Deaths and serious injuries remain all too common in the construction industry, with nearly one worker a week losing their lives; this is a real problem, rather than a theoretical issue.
“The question of whether all workers on a site will be enrolled onto the B&CE scheme should be one of the first asked when clients are awarding contacts, companies that answer no should simply not be allowed to operate in construction.
“The failure to enrol workers onto the scheme is unacceptable to Unite and must become unacceptable to the industry in the same way that having the correct PPE (personal protective equipment) is.”