The notion that Brexit could save the NHS by freeing the health service of burdensome immigration is a “dangerous lie” that could see the UK starved of medical resources, four former Labour health secretaries have warned.

Alan Milburn, Patricia Hewitt, Andy Burnham and Alan Johnson said leaving the EU would risk “frightening consequences for staffing, waiting times and levels of service care”.

Their statement, which comes as the future of the NHS emerged as a key battleground in the EU debate, supports the warning by Jeremy Hunt, the current health secretary, that a vote to leave would put NHS funding at risk.

Campaigners for Brexit have argued that leaving the EU could secure the health service, reducing the burden on the NHS caused by immigration from Europe. Priti Patel, the employment minister, has warned the health service is under threat because of the UK’s membership, an argument echoed by Michael Howard, the former Conservative leader.

“We will be talking a lot about the NHS in this campaign because we believe that a leave vote is vital if we are to protect the NHS for future generations,” Patel said. “Current levels of migration are causing unsustainable pressures on our public services and we can see that the NHS is creaking under the strain.”

The former health secretaries rebutted this idea in their letter, saying the economic implications of Britain leaving Europe would be “felt at the heart of our cherished national institution”.

“There is a dangerous lie being propagated by those wanting to leave the EU that they will protect the NHS. The opposite is true,” they said.

“These are the same people who have campaigned for increased NHS charging, increased privatisation and cuts to spending. People should not trust their motives or their maths.”

The Labour politicians argued leaving Europe would mean less, not more, money for the NHS: “Independent experts have repeatedly shown that leaving will seriously damage our economy with some suggesting Britain would enter recession. That means reduced funding for public services – strangling NHS finances, with potentially frightening consequences for staffing, waiting times and levels of service care.

“The economic damage wreaked by leaving Europe will have devastating knock-on effects, including in the health service. Given the millions of lives who depend on our NHS, it is a risk we simply cannot afford to take.”

The row comes as new research suggested the leave campaign is gaining ground and may benefit from higher turnout among its supporters.

Research by the Fabian Society found remain had a narrow lead among theadult population, but highlighted three factors in favour of the leave campaign: leave voters being more likely to vote, leave arguments seeming to have a greater power to persuade and weaker loyalty to their cause among remain’s voters.

An online Opinium survey for the Observer found the Brexit campaign on 43% to remain’s 39%, with only just over half of under-35s planning to vote compared with more than 80% of over-55s.

Ken Clarke, the Conservative former cabinet minister and pro-EU campaigner, said he was worried about the findings.

“That poll is very worrying because it attributes that to the fact that young people have not been engaged in it and are unlikely to vote,” he told Radio 5 Live’s Pienaar’s Politics. “It’s them who are going to be living with the consequences of the vote that seem least inclined to turn out to vote.

“I can’t think of a more disastrous way of resolving the future of the country. Of course the short-term effects of having Brexit in the present state of the global economy, the European economy, the British economy would be catastrophic.”