Calls have also been made for a legal change to automatically protect the rights of EU citizens and their families who continue to live in the UK, while concerns have been raised there will be thousands of undocumented EU children who have not had settlement applications yet made on their behalf by parents or social workers.
The EU Settlement Scheme (EUSS) was established to allow EU citizens and their families to apply to the Home Office to confirm their immigration status to continue living and working in the UK. More than 5.6 million applications were made.
However, there is believed to be a backlog of around 400,000 cases awaiting a decision, and despite calls for an extension amid delays and difficulties during the coronavirus pandemic, the UK Government has insisted this will not happen.
Last week, the French Government extended the deadline for its post-Brexit residency scheme by three months.
Scottish Labour MP Ian Murray said the “intransigence” of the UK Government could see EU citizens living and working legally in Britain fall foul of immigration laws because the deadline was missed as a result of administration issues.
“It is vital that European Union citizens across the country register for the settlement scheme today if they haven’t already,” he said.
“But with hundreds of thousands of applications still unprocessed and potentially many more yet to apply, the deadline must be extended beyond June 30.
“The home secretary must reconsider and join European governments who have extended their deadline for UK nationals living in the European Union.”
SNP MP Kirsten Oswald said: “This is the Prime Minister’s last chance to prevent hundreds of thousands of EU nationals being stripped of their rights overnight, as well as a hammer blow to Scotland’s economy and NHS.
“I am urging him to take it and scrap the deadline.”
Ms Oswald said the government should “implement a declaratory system whereby EU nationals in the UK are automatically granted settled status”.
Meanwhile, the Children’s Society has estimated that thousands of EU children living in the UK will become undocumented this week because applications to the EU Settlement Scheme have not yet been made on their behalf, while others are caught in a growing backlog of cases.
And Maike Bohn, co-founder of campaign group the3million, which seeks to protect the rights of EU citizens living in the UK, said it would have been “absolutely right” to extend the deadline by a year because of the coronavirus pandemic.
But she said now the government needed to "tweak" legislation to "make people legal from the 1st of July who can get status successfully later and that way they have protected everyone".
A Home Office spokeswoman said: "If someone hasn't applied by the deadline of June 30, we will take a flexible and pragmatic approach.
"They will be encouraged to apply as soon as possible and we have been clear that where someone has reasonable grounds for missing the deadline, we will work with them first to give them the opportunity to apply.”