The proposal is part of a series put forward by the Institute of Public Policy Research to help the UK manage the threat of EU citizens leaving.
As well as offering automatic citizenship to NHS workers, the UK should offer citizenship to European children educated in Britain, the think-tank says.
It also calls for EU citizens resident in the UK to be given indefinite leave.
Researcher Chris Murray said: “It is critical to public health that these workers do not seek jobs elsewhere. All EU nationals who work for the NHS, or as locums in the NHS system, should be eligible to apply for British citizenship.
"This offer should be organised by the regional NHS and mental health trusts, who would be responsible for writing to all NHS staff who are EU nationals to inform them of their eligibility.”
He added: “There are currently around 57,000 EU nationals working in the English NHS, accounting for 5% of its workforce. One in ten of the UK’s registered doctors is an EU national. Without them the NHS would collapse.”
Labour deputy leader Tom Watson wrote to the Prime Minister yesterday following a series of leaks of internal government documents.
He has called for a proper evaluation of the need for extra weekend working, suggesting it is conducted by the National Institute for Health and Care Excellence.
His call came after a risk register was leaked showing the lack of staff for extra weekend working as a major threat to the seven-day project.
A second leak showed that Health Secretary Jeremy Hunt has been advised to stop talking about the so-called "weekend effect."
Mr Watson said an independent inquiry should establish whether there would be any clinical benefits for patients from the seven-day policy and whether it was a cost-effective use of NHS resources.
In his letter to Mrs May he states: “The seven-day NHS policy has nonetheless been used to justify imposition of a new contract upon junior doctors.
“That is wholly unacceptable, given the current paucity of evidence underpinning the policy.
“Imposition should be suspended and doctors allowed to remain on their current contract until such time as you can demonstrate the evidence justifying the policy.”
Instead the appointment for the head of the Yorkshire and Humber School of Radiology is to be rerun, according to Health Education England.
The appointment provoked strong opposition from the Royal College of Radiologists, which said it had not been involved in the appointment process.
Health Education England says the college is to be involved in the new appointment.
Its medical director Professor Wendy Reid said: "It is clear that while our internal processes were followed and the candidate appointed is of the highest calibre, HEE should have involved the college from the very start of this appointment process.
"We have agreed that the best course of action now is to rerun the appointment process, in collaboration with the college, providing them, the wider profession, and local trainees the assurances that they require.
"HEE is clear that multi-professional learning can deliver the best learning environments and remains committed to making the best appointment for all our Heads of Schools, recognising that different skills and knowledge can deliver excellent education and training to meet trainees’ and patients’ need in a modern NHS.
"I will be writing to the college shortly to establish the next steps."
The department at Grantham Hospital, Lincolnshire, is currently closed overnight - and closure is expected to last three months.
But a campaign group called SOS Grantham Hospital has recruited a leading law firm, Leigh Day, to make its case.
The lawyers say the closure decision was unlawful because there was no proper consultation with patients.
Rosa Curling, from the company, told the Health Service Journal: “It is our opinion that the decision to close Grantham Hospital overnight for the next three months is unlawful. It was taken without proper patient involvement. Any decision to close an A&E department at a popular local hospital is a serious decision that clearly requires a proper consultation process.
“The people of Grantham are being denied vital services and we are now representing the SOS Grantham Hospital group in their fight so that the people of Grantham have a strong voice in the NHS decision making process."
United Lincolnshire Hospitals Trust medical director Suneil Kapadia said: “Due to a severe shortage of doctors, we had to make this difficult decision quickly, so it hasn’t been a perfect process. We didn’t act unilaterally and worked with NHS partners and other stakeholders where possible.
“As the closure is temporary and made quickly on the grounds of patient safety, unfortunately we were not able to consult the public."