The government has been urged by MPs to urgently change its policy on EU citizens in the country if it is to avert a “Windrush-style catastrophe” in the years after Brexit.

Politicians on the influential home affairs select committee said they had serious concerns about the design of the settlement scheme for EU citizens, launched by the Home Office two months ago.

There are an estimated 3.8 million EU citizens in the UK and those who want to remain after Brexit are obliged to apply for settled status by June 2021 in the event of a deal or December 2020 in the event of no deal.

Committee members said the design of the scheme meant many EU citizens were at risk of forfeiting their rights to remain after the deadline. “The prospect of a Windrush-style catastrophe happening to over 3 million EU citizens who have made the UK their home in good faith is deeply troubling,” the committee said in a report, EU Settlement Scheme.

The report comes as new statistics show 750,000 people have applied for settled status since the scheme opened, among them 100,000 of the estimated 1 million Polish people in the UK.

The home secretary, Sajid Javid, said the numbers were “immensely encouraging”.

After an investigation and witnesses including Javid, the committee’s report concluded the only way to ensure EU citizens are guaranteed to retain their rights is to legislate.

“We therefore call on the government to protect in law the rights of EU citizens in the UK. The government should guarantee in law that any EU citizens living in the UK before Brexit are legal residents of the UK and are able to continue to live and work as they have done until now,” says the report.

The Labour MP Yvette Cooper, who chairs the committee, said: “The government’s current plans for the EU settlement scheme show they are not learning the lessons from the Windrush scandal.

“The problems faced by the Windrush generation showed how easily individuals can fall through gaps in the system through no fault of their own and how easily lives can be destroyed if the government gets this wrong.”

Stuart McDonald, the SNP member of the committee, said that under the scheme “too many people, including children and vulnerable individuals, risk falling through the gaps and not accessing the scheme at all … The warning signs are there, now the government must take action.”

He also said the government needed a printed document and not a digital system to enable EU citizens to deal with landlords, employers and officials at airports and ports.

“People also need hard copy documents, not just an unfamiliar digital system,” he said.

Echoing the criticisms of campaigners, the committee said EU citizens should not have to apply to remain but just have to declare they are in the country.

This would bring the approach into line with other EU countries, where people are merely required to notify a town hall of their arrival and their address.

“The government has chosen to implement a system which does not grant status to eligible people but requires them to apply for it and the home secretary told us that EU citizens are only entitled to the status which they are able to evidence.

“We disagree with this. We believe the EU citizens legally resident in the UK [before Brexit] should have their rights protected and their entitlement to remain enshrined in law,” says the report.

Witnesses told the report “there is another way forward” similar to the treatment of Commonwealth citizens who had their immigration status formalised in the 1970s.

As Colin Yeo, an immigration barrister, told them: “You just pass a law saying ‘You are lawful. We will sort out the difficulties later, as and when they arise.’”

The issue over the government approach to EU citizens came to the fore earlier this week when Michael Gove pledged to support calls by Tory backbencher Alberto Costa for a declaratory system. He has also offered free British citizenship to EU citizens.

Ministers faced a furious backlash over the treatment of the Windrush generation after it emerged long-term UK residents were denied access to cancer treatment and other services, held in detention or removed, despite living legally in the country for decades.

A Home Office spokesman said: “We disagree with the home affairs select committee’s assessment of the scheme, which is performing well with more than 600,000 applications received by the end of April and hundreds of thousands of people already being granted status.

“The scheme protects the rights of EU citizens in UK law and gives them a secure digital status which, unlike a physical document, cannot be lost, stolen or tampered with. A declaratory system – that means EU citizens are not required to obtain status and evidence of this – risks causing confusion especially for the most vulnerable, and could in years to come find people struggling to prove their status.

“We have taken great care to learn from the experience of the Windrush generation. It’s part of the reason why there are 200 assisted digital locations across the UK to help EU citizens apply, dedicated staff in our Settlement Resolution Centre and £9m available for 57 organisations across the UK to support an estimated 200,000 vulnerable people to apply.”