Alberto Costa, the MP for South Leicestershire, told Sky News large numbers of those who need to apply are elderly or vulnerable and should not be forced to go through the procedure.
The settled status scheme fully launches today, with a multimillion-pound advertising campaign aimed at EU nationals.
But campaigners say people who have signed up during a test phase, which has been open for several months, have struggled to provide the correct documents even if they meet the requirements.
Mr Costa is calling on the home secretary to make it “declaratory” so people do not need to sign up except in certain situations, such as applying for a job or renting a property – and do not lose their rights.
He also believes those who have lived here for five years should be automatically offered British citizenship.
Citizens of EU countries, as well as Norway, Iceland and Switzerland, who can provide evidence of residence in the UK for five years, will need to apply for settled status by the deadline of 31 December 2020.
It may be extended until June 2021 if there is a deal with the EU. Those who have lived here for less than five years can apply for “pre-settled” status, which they can convert to settled status after five years.
If Britain leaves the EU without a deal, they must be living in the UK by 12 April.
Mr Costa was forced to resign as a junior government aide this year for making proposals in parliament to protect EU citizens if Britain leaves the bloc without a deal.
He is now urging Home Secretary Sajid Javid to make changes to the settled status scheme.
Mr Costa said: “Windrush still looms very large over the Conservative government. We have 3.6 million EU national residents in the United Kingdom and we have to be absolutely clear that we want to avoid any possibility of a Windrush-type scenario.
“I think it’s profoundly un-British and frankly immoral to be asking people to apply for rights that they’ve exercised in many cases for decades, many of whom are elderly, many of whom are vulnerable, many of whom are in care homes, and don’t know that the settled status scheme applies to them.
“That’s why I believe there should be a declaratory approach. So you keep those rights, and the registration is only to get a document to confirm those rights for example if you need to show a document to rent a property or for employment.
“The outcome of current government policy is that if you don’t register by 2021 you don’t acquire those rights. That means you become an illegal alien overnight.
“I am in discussions with the home secretary to suggest that a way of encouraging people to register by 2021 would be perhaps to offer them British citizenship as a gesture of goodwill by the British state to EU nationals.”
The Home Office say the system, which is accessed on a mobile phone app, is free of charge and simple to use.
People can apply online with proof of their identity, their residence in the UK and any criminal convictions.
Irish nationals and people already granted indefinite leave to remain do not need to apply.
Around 50 centres have opened across the country today for people struggling to use the app to scan their documents.
Hildegard de Souza, 81, moved to London nearly 60 years ago from Germany, and has been here ever since after meeting her British husband.
Now a widow, she told Sky News she struggled to apply for settled status after she was asked to provide additional proof of residence.
She has now been approved after receiving help from the Citizens Advice Bureau through her local council.
Mrs de Souza said: “For people of my age and not experienced with the internet, it’s impossible to do it. It caused me quite a lot of anxiety.
“I realised that I didn’t have my passport that I landed with in 1961. I worked and claimed social security and when I had children, claimed child benefit and things like that. It was quite an awakening to find I couldn’t produce the documents.
“Luckily I had some help but I don’t think everybody does. Can everybody afford a solicitor to do it? I doubt it.”
Campaigners say women are disproportionately affected as they are more likely to lack a continuous work history, and many have encountered problems with documents in their married surname.
A third of female EU nationals in the UK are aged over 75.
Monique Hawkins is a spokeswoman for “the3million” campaign, which is working for all EU citizens to retain their existing rights after Brexit.
She told Sky News: “We’ve seen such nightmares.
“People’s lives are complicated and this system is just going to create hundreds of thousands of problems.”
Ms Hawkins said in the event of no deal, EU citizens will also have fewer rights to appeal if they are rejected.
She said that £9 million put up by the Home Office for charities to help vulnerable people is a “drop in the ocean”, and added: “A lot of countries are looking now at what the UK is doing and wanting to implement similar schemes, which mean you don’t have rights until you apply.
“They’re looking closely at what the UK is doing.”
British citizens living in Europe have told Sky News they have received minimal information about their rights after Brexit, with reports that individual cities and regions are making their own rules.
Hellen Mallaburn, 43, and her husband Duncan who moved to the French Alps 11 years ago.
They run a catering company and said despite working and paying taxes throughout their time there, they have only received a one-year temporary residence card – which they are appealing.
Ms Mallaburn, originally from South Yorkshire, said: “It’s quite uncertain. France have put in place procedures for us if there is no deal but they have stated quite clearly that whatever the UK decides to do it will be reciprocal for us.
“It makes me very concerned, seeing the difficulties citizens are having in the UK that we’re going to be facing the same or similar and our voices are not being heard.”
Immigration minister Caroline Nokes defended the scheme and said £9 million would be given to charities to help vulnerable people access it.
She said: “The point of this is to enable EU citizens who are living here, who are working with us, who are contributing to our communities and to our economy to be able to evidence their status, to evidence their right to access services and benefits. We know these people are here legally and we want them to stay.
“It’s really important to me that we don’t repeat the mistakes that were made with Windrush.”