Tens of thousands of EU citizens living in the UK will be issued with a formal 28-day notice if they have failed to apply for post-Brexit settled status within a week, the government has warned.

The notices will tell them to submit an application or risk consequences which include losing their rights to healthcare and employment.

With a week to go before the Wednesday 30 June deadline for the EU settlement scheme, the Home Office is redoubling efforts to reach those who do not know about the rule change, including vulnerable groups such as elderly people and children in care.

Immigration minister Kevin Foster said it had ruled out extending the deadline despite a huge surge in applications, now running at 10,000 to 12,000 a day.

“Put simply, extending the deadline is not the solution to reaching those people who have not yet applied, and we would just be in a position further down the line where we would be asked to extend again, creating more uncertainties,” he said.

However, he said EU citizens who had had failed to apply by the deadline would not see their social welfare benefits cut off from 1 July and promised the Home Office would be flexible and lenient.

The Home Office has had 5.6m settled status applications from EU citizens including some repeats. Officials revealed a backlog of 400,000 which may take until the end of the summer to process.

It has received 1.5m helpline calls and 500,000 requests for help through an online contact form – an indication of the unprecedented scale of the exercise and the challenges many are facing.

To accommodate those who will not have a decision for months, the government will issue a “certificate of application” which all applicants can “rely on as proof to access their right to work or rent”, Foster said. It can also be used to access the NHS. “People will not lose their benefits next week,” Foster told members of a House of Lords committee.

In a briefing to reporters on Tuesday, the Home Office said it would work with individuals to find out their reasons for not applying rather than deport them.

“We’ll set up the support available and we’ll signpost people to make an application, but we do recognise that there may be some people who, after that 28 days, still haven’t been able to make an application, and then I think we would want to work with them to understand why that is the case, and then support them again to make the application,” said an official.

The flexible approach is not expected to be permanent, however, and under immigration rules EU citizens who make late applications will have to provide “reasonable grounds” for not applying. Among those set to be issued with a 28-day notice are EU citizens unable to evidence their right to work, who might be identified by immigration enforcement teams making checks on employers.

The Home Office said the NHS will continue to be available to those who have certificate of application and “urgent treatment” would never be refused.

It warned that employers or landlords do not have to do retrospective checks on their workers and those who reject future workers or tenants on the grounds that they have not yet been granted official post-Brexit status can be sued on discrimination grounds.

The new rules were brought in as a result of the UK’s departure from the EU and are designed to protect those already living in the UK and British citizens already settled in the EU when Brexit came into force on 1 January this year.

The Home Office said it was concerned that British nationals were facing issues in relation to reciprocal residency rights in some EU countries including Bulgaria, Italy and Portugal and had raised this with the EU ambassador on Tuesday.

Part of the government’s policy is driven by the Brexit pledge to “take back control” of the borders and reduce immigration.