The Home Office said so far just two applications have been refused on "suitability grounds".
But analysis by the PA news agency of the department's figures indicate around 7,600 applications did not result in people being granted full permanent permission to live and work in the UK - called settled status - or temporary leave to remain, known as pre-settled status.
The news prompted concerns from campaigners and immigration lawyers who fear EU citizens and their families could be put at risk of deportation if they are not granted leave to remain before the deadline.
They claim the publication of "select figures" is "disingenuous" and could give a "misleading" impression that the scheme is successful.
EU citizens and their relatives - including those from the European Economic Area (EEA) countries of Iceland, Liechtenstein and Norway as well as Switzerland - have been asked to apply to confirm their immigration status so they can live and work in the UK when freedom of movement ends.
Relatives of EEA and Swiss citizens who are not from any of those countries but all currently live in the UK under EU law are also being urged to apply.
Once granted status, applicants can use the NHS, study and access public funds and benefits, as well as travel in and out of the country. But first they must prove their identity, show they live in the UK and declare any criminal convictions before a December 2020 deadline.
The total number of applications finalised by the end of September was more than 1.5 million (1,524,500).
Of these, 61% were granted settled status and 38% were given pre-settled status - which can be applied to be updated once someone has lived in the country continuously for five years - while the conclusion in 0.5% of cases was classed as having "other outcomes".
The data says cases with other outcomes include applications deemed: "invalid" as mandatory information was not provided; "void" because the applicant was ineligible; were "refused on eligibility or suitability grounds"; or had been withdrawn by the applicant.
Incomplete applications or those from British citizens fall into this category.
All the statistics published were branded experimental and should be "treated with caution", the department warned, adding that percentages may not add up to 100 due to rounding of numbers.
Maike Bohn, co-founder of campaign group the3million, told PA: "The figures are disingenuous and self-serving as they are trying show the scheme is going well.
"The Government is hiding behind formalities and soundbites. The devil is in the detail."
She claimed the Home Office could be "hiding refusals" by recording decisions which ultimately constituted this under different headings, adding: "It is very select data so we cannot clearly know."
Zeena Luchowa, an immigration solicitor at Laura Devine Law, told PA large numbers of people were seeking legal advice on the scheme because they had found the process confusing, adding: "There's a real concern as to what exactly is going on at the Home Office and the types of decisions that are being made.
"What is worrying is trying to find out what is happening when there are rejections.
"Being rejected doesn't mean the application has been refused but it still has ramifications."
She said the scheme did not appear to be as straightforward as intended.
The Home Office, which refused to provide a full breakdown of the figures when asked, said the scheme was "performing very well" with 20,000 applications being processed a day, initially adding: "Our statistics are clear - two people have been refused - and it's complete nonsense to suggest otherwise."
The department later amended its response, to instead say it was "completely untrue" to suggest otherwise.
Every application was "reviewed by a caseworker who will always contact the applicant to seek to resolve any issue", there was a helpline to call and no-one would be granted pre-settled status "without first being offered the opportunity to submit evidence that they are eligible for settled status," a spokesman said.
More than 1.8 million people had applied by the end of September, according to the statistics. But the latest internal figures show a total of two million applications, the Home Office said.
There were an estimated 3.6 million EU citizens living in the UK in 2018, but there could be nearer four million now, data and research by immigration experts indicates.
In August, the Office for National Statistics admitted it had underestimated EU net migration to the UK and the figures could be higher than first thought.
Mayor of London Sadiq Khan called on the Government to "urgently do more", saying it was "hugely concerning" around two million EU citizens living in the UK and their families were yet to apply for the scheme.
Earlier this year, the scheme came under fire when chef Damian Wawrzyniak, who has twice cooked for the royal family, was denied settled status despite saying he provided proof of living in the UK for almost 15 years.
Last week, Yvette Cooper, chairman of the Commons Home Affairs Committee, asked Home Secretary Priti Patel to provide "accurate information" because there was "still a lack of clarity" over some aspects of the scheme.
The Advertising Standards Authority (ASA) banned a radio advert for the scheme for failing to make clear documents as well as a passport or ID card would be needed to apply - a decision the Home Office said it completely disagreed with.