Edvardas Zubavicius, 33, who lives in Hull, said he had to travel back to his home country for a passport after the government’s EU Settlement Scheme app declined his Lithuanian ID card.
The Home Office, which runs the scheme, said in a statement any applicant struggling to get documents verified could send them to officials through the post.
It comes as Liberal Democrat Coun Mark Ieronimo warned of “another Windrush” if EU citizens miss the Wednesday, June 30 deadline and called on Hull City Council to make them aware.
A council spokesperson said it was publicising the national scheme to EU nationals through its area teams and sent them a letter with last year’s council tax bills.
Minister for Future Borders and Immigration Kevin Foster said the Home Office had launched a recent campaign urging EU nationals to apply.
Mr Zubavicius, of Welwyn Park Road, who has lived in Hull for eight years, said: “My family and I first heard about the settlement scheme on the news before we got any letters, I spoke to my mum and brother and we shared information between us.
“So we knew we would have to apply though at first we were uncertain as to how. I understand it would have been hard putting the system in place.
“It was a new thing and the UK was the first country to leave the EU so there was no example of how another country had dealt with it, we knew there’d be some problems and we accepted that.
“The government app said it accepts national identity documents but I tried it multiple times and it didn’t work. Lithuania is in the Schengen area so I’d never needed a passport before, I don’t know why the app didn’t accept it as it has basically the same information.
“At first I tried to get a passport from the Lithuanian embassy, that was fully booked for the next month or two so I had to go back to the country. Once I got the passport there were no problems at all, the process was easy and the app accepted it.
“I’ve lived here for eight years, when someone asks me about home I think of Hull.”
Coun Ieronimo said that although Mr Zubavicius was eventually able to successfully apply, the current system could see others “fall through the net”.
The councillor said: “The one’s were not reaching seem to be those who have been here a long time, they’re adapted to the British way of life and many of them don’t have a home in their origin countries anymore, or even speak the language.
“It’s not quite a cliff edge, there will be exceptions to the deadline. I think the council could be more vocal on this, I’d like to see a clear message from the top saying to EU citizens you’re welcome here and we want to make sure you can still access things like opening bank accounts, the things they’re entitled to.
“They could even look at doing basic things like put up posters around the city in different European languages. We don’t want another Windrush if EU citizens aren’t able to apply before the deadline.”
A Hull City Council spokesperson said: “The EU Settlement Scheme is a national Home Office initiative designed to help EU citizens all over the UK to apply for settled status before the Wednesday, June 30 deadline.
“Hull City Council works with the Government to share these Home Office messages to EU citizens in Hull, while the council’s Area Teams are working to communicate the details of the scheme to neighbourhoods across the city.
“The council also included a letter containing information about the scheme to each Council Tax payer with their bills last year, sends regular guidance to all businesses in the city and offers additional support through its Customer Service teams.”
Future Borders and Immigration Minister Mr Foster said: “Every day, thousands of people are being granted status under the hugely successful EU Settlement Scheme with more than 4.9 million grants of status made already.
“I would urge people who are eligible to apply as soon as possible to secure the status they deserve in UK law, with a range of support available to help them do so.
“We have launched a new campaign to encourage those who need to apply to act before the Wednesday, June 30 deadline and for anyone struggling with their application our EU Settlement Resolution Centre is available to talk them through their application and what they need to apply.”
The Home Office said in a statement EU citizens could use the EU Exit: Identity Check app to verify documents.
They added officials would work with applicants to confirm residency based on a range of evidence, including bank statements, pay slips and employer letters.