They fear the EU is poised to weaken its stance on the issue at a make-or-break summit next month in order to keep negotiations moving between London and Brussels.
Controversial issues including the future rights of EU citizens to bring in family members, the rights of their children born after Brexit, and compulsory new criminal record checks.
EU nationals in Britain say new “settled status” rules will reduce their rights and be based on UK immigration law – exposing them to the Home Office’s self-described “hostile environment” towards immigrants.
A Government minister recently admitted EU citizens could be deported before being able to challenge any rulings that they are not entitled to stay after Brexit.
Key issues will be “shunted” into phase two of the talks, rather than being settled before the negotiations move onto future trade terms between the UK and the EU, MEPs and citizens fear.
They spoke with The Independent after meetings with Michel Barnier, the EU chief negotiator, who will recommend whether “sufficient progress” has been made to progress the talks.
It comes as the Prime Minister prepares to hold a crucial lunch meeting with Jean-Claude Juncker, the Commission President, and Mr Barnier in Brussels on Monday.
Before meeting Ms May, however, the pair will hold talks with Guy Verhofstadt, the European Parliament’s Brexit coordinator, to discuss progress in the negotiations.
In recent weeks, the two sides have moved closer to agreement on both a financial settlement and the Irish border – with less focus on citizens’ rights, the third “divorce” issue.
Seb Dance, a Labour MEP and a member of the Parliament’s EU Citizens’ Taskforce, said: “Michel Barnier did not go far enough. There are concerns that the EU is ready to dilute its stance on citizens’ rights – but we won’t sit back and let that happen.
“Family reunification is one of the areas that the EU may be ready to compromise on.
“We can’t move on just for the sake of moving on. We should only move on when we are certain that nobody will suffer as a result because of a decision they had no part in.”
Jean Lambert, a Green MEP and taskforce member, agreed, saying: “It looks as if issues may be shunted into the second round of negotiations.
“The issue of family unification is crucial. We can’t have children born after Brexit with different legal status, that’s not going to be satisfactory.”
Ms Lambert said the ECJ “must have a role that runs through these people’s lives”, adding: “We want to see these things cleared up in the first stage of the negotiations
“I will be really worried if these things are taken off the table for now. I would like to see the EU being more robust on it now.”
Meanwhile, British In Europe, an alliance of groups representing more than a million British citizens living in the EU, have written to European leaders urging them to block Brexit negotiations from moving forward.
“We might be told we are closing in on a deal but not one that is good for the people who will have to live with it,” a spokesperson for the group told The Independent.
The group also warned the issue could “get bumped into phase two of the negotiations, running the risk of being lost in the debate”.
And Anne-Laure Donskoy, co-chair of the 3milliongroup, representing EU citizens in the UK, said: “There is a lot of pressure on both sides of the Channel to move to phase two.
“When Ireland has the peace process and economic ties as pressure points and the UK the divorce settlement bill, we “only” have the moral argument.”
Both MEPs and campaigners fear any agreement now will be lost if the negotiations collapse later and Britain crashes out of the EU with no deal.
For that reason, the Commons Brexit Committee, on Friday, called for citizens’ rights to be “ring-fenced” from the rest of the talks, to ensure an agreement is secured.
The EU’s original guidelines for the Brexit talks vowed that its “first priority” would be to “safeguard the status and rights” of EU citizens, something Mr Barnier has repeated after various rounds.
However, he has recently spoken of how progress over citizens’ rights has moved the two sides “closer to a deal”, while the Prime Minister said they are within “touching distance”.
On the ECJ, Britain floated an idea that British judges could voluntarily refer cases to the European court if legal queries arose, but is believed to have backtracked.
Mr Barnier will also come under pressure to ensure citizens’ rights are fully protected when he meets Frances O’Grady, the TUC General Secretary, next week.
“There is nothing stopping Theresa May making a unilateral guarantee of the right to remain for EU citizens who have made their life in the UK,” Ms O’Grady told The Independent.
“It would create goodwill and clear the table for negotiations to focus on protecting jobs and rights at work.”
Ms May will hope to reach agreement on progressing the talks as early as Monday, when she meets Jean-Claude Juncker, the European Commission President, in Brussels.
If “sufficient progress” is not made next month, the next opportunity will be in March – by which time UK businesses have threatened to implement plans to move some of their operations to the EU.
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