MPs have urged the Prime Minister to introduce an "NHS guarantee" while the Liberal Democrats have made calls for the Government to implement an "NHS passport" to protect the rights of health workers from the continent.
Tory former health minister Dan Poulter said it was essential not only to reassure existing NHS workers but also allow the health service to continue recruiting from the EU.
The MP, who also works as an NHS doctor, told The Observer: "The NHS is heavily reliant on the contributions made by many dedicated EU healthcare professionals, and without them, our health and care system simply wouldn't be able to cope."
He added: "Protecting the rights of EU nationals to continue to live in the UK and care for patients is essential, but it is also important that we look after the best interests of the patients of tomorrow.
"Having the right work visa rules to ensure that in future, healthcare professionals from within the EU can continue to contribute to the NHS and care for patients, must be a priority."
The newspaper said a total of 2,348 doctors from the 27 other EU states left NHS England between July and September 2016 compared with 1,281 in the same period in 2015 - a rise of 83 per cent.
Shadow health secretary Jon Ashworth said: "This NHS crisis will be compounded if the 140,000 EU nationals working in the NHS and social care sector walk away.
"It's time for an NHS guarantee for these workers ensuring their rights - offering these workers who care for our sick and elderly the certainty that they deserve.
"What's more, we need urgent guarantees that the NHS will be able to continue recruiting from the EU as it currently does. Yet we have no clarity whatsoever from the Prime Minister."
A Government spokeswoman said: "Overseas workers form a crucial part of our NHS and we value their contribution immensely.
"The Government has been clear that public services and other industries that depend on migrant labour would need to be able to continue to recruit workers from overseas."