Britain’s departure from the EU would hit housebuilders’ workforces hard and exacerbate the housing crisis, the boss of the UK’s largest homebuilder has said.

David Thomas, chief executive of Barratt Developments, said: “We would much prefer that the UK stays within the EU. We have a significant part of our labour force, particularly within the London market, coming from continental Europe – the free movement of labour in the European market is a positive from our point of view.”

A vote to leave the EU would mean “even more pressure in terms of skills shortages”, he argued. “If you ask any housebuilder what their main challenge is, they say it’s labour availability.”

This would hamper the industry’s ability to build much-needed homes, he said, echoing warnings from other housebuilders, including Berkeley Group.

Thomas estimates 30-40% of Barratt’s workforce in the capital hails from mainland Europe. “It wouldn’t be unusual to find 10-plus nationalities on a London construction site, and dual language signs,” he said.

Two-thirds of the construction sector is backing continued EU membership, with more than half saying Brexit would lead to less foreign investment (60%) and drive up labour and material costs (55% and 53%), according to a survey by Building magazine.

The Shard contractor Mace, the property firm Jones Lang LaSalle, and the UK’s largest independent engineering firm, Mott MacDonald, have signed an open letter backing the vote to remain.

Mott MacDonald’s chairman, Keith Howells, said: “We would face quite a significant skills shortage if we opt out [of the EU]. We employ quite a number of EU nationals. A lot of young people have come here from Greece, Spain and Italy, got masters degrees and put themselves on the local market. What’s the impact going to be on them? We’re all in the dark.”

At the property developer Berkeley, which builds luxury London homes, about half of its 14,000 subcontractors come from eastern Europe, according to the chairman, Tony Pidgley.

Across the UK, nearly 12% of the 2.1 million construction workers come from abroad, official figures show – mainly from the EU. Experts say the actual number is probably even higher. Poland and Romania are the most common countries of origin.

The construction industry employs 324,000 fewer workers than it did in 2008, before the financial crisis and recession led to a slump in housebuilding and other construction projects, prompting companies to slash their workforces. This has made the sector heavily reliant on construction workers from eastern Europe and the rest of the EU, as it takes years to train skilled tradespeople.

The number of migrant workers more than doubled in the 10 years to 2011, said Simon Light at design and consultancy firm Arcadis. “The migrant workforce has been the most effective short-term fix for the UK’s construction worker shortage.”

A report by his firm last year found that 53,000 extra bricklayers were needed to build 200,000 homes a year. But the UK is also recruiting carpenters, joiners, plasterers and general labourers along with architects from the EU.

Steve Turner of the Homebuilders Federation said: “Because the industry is reliant on labour from the EU, in the event of Brexit we would be pushing the government hard for a quota system … Considering that housing is such a priority for the government, we would anticipate that the government would listen to our concerns.”

He pointed to a quota system for construction workers from Ireland in the 1960s. Thomas warned that the uncertainty created by a vote to leave the EU in the 23 June referendum would also hurt investment, with housebuilders operating on a five- to 10-year basis. Barratt has invested more than £3.8bn in land for new housing in the past five years.

Its business has been little affected by the forthcoming EU vote, apart from the top end – reservations of £1m-plus homes have slowed slightly, to 44 in the past 19 weeks, compared with 46 a year earlier.

Overall, forward sales rose nearly 10% to £2.8bn, underpinned by the government’s latest help-to-buy scheme, which started in February.

Savills, the upmarket estate agent, said there was a spike in activity in its UK prime residential business before the stamp duty increase last month, but it had slowed since then in the run-up to the referendum.

Migrant construction workers in UK by nationality

The most frequent nationalities of people resident in the UK and working in construction in 2014 – these people have arrived in the UK in the past 10 years, from figures by the Construction Industry Training Board:

Poland: 30,120
Romania: 24,842
Lithuania: 7,569
India: 7,704
Bulgaria: 5,443
Latvia: 3,830
South Africa: 1,316
Hungary: 1,448
Australia: 937