Hungary, Poland, Czechia and Slovakia have said they're ready to veto any Brexit deal that would limit their citizens' rights to work in the UK. Article 50, which will begin the process, has yet to be triggered.
In an interview with Reuters on Saturday, Slovak Prime Minister Robert Fico said that having already opposed the EU's efforts to introduce mandatory quotas for migrants, the Visegrad Group (V4) of Central European countries would extend its common interest in protecting citizens' rights to work in the United Kingdom.
The comments followed a meeting between EU leaders in the Slovak capital Bratislava on Friday, which marked the first summit in decades that was held without the UK being present.
In an EU referendum on June 23, 52 percent of the UK's eligible electorate voted in favor of leaving the 28-member bloc.
May has said reducing immigration into the UK is crucial to the final Brexit deal
Almost three months since the Brexit vote, the EU is still waiting on the UK to trigger Article 50 - the formal step required to begin the departure of a member state from the bloc.
But British Prime Minster Theresa May has said the divorce proceedings won't begin until 2017, giving her government time to prepare for negotiations.
After playing a major role in the "Vote Leave" campaign, May has said that reducing immigration into the UK is crucial to the final Brexit deal. EU leaders have warned, however, that membership in the single market is conditional on Britain accepting the free movement of people.
Fico said Saturday that the V4 countries will be "uncompromising" in the deal, "unless we feel a guarantee that these people [living and working in Britain] are equal, we will veto any agreement between the EU and Britain."
"I think Britain knows this is an issue for us where there's no room for compromise," Fico added, reiterating that he was opposed to any "cherry-picking" in negotiations.
The Brexit vote has triggered what European Commission President Jean-Claude Juncker has described as an "existential crisis" for the EU.
Friday's informal meeting of the bloc's remaining 27 members intended on displaying unity but instead highlighted the Union's divisions over migrant policy.
Since more than 1 million migrants fled to Europe from war and poverty-stricken countries in Africa and the Middle East last year, the EU's eastern members have been at odds with the bloc's older members, particularly Germany, which has received its fair share of criticism and praise for its open-door policy.
Slovakia has been one of the harshest critics of quotas and has sued the bloc over a plan agreed last year to redistribute migrants, but was outvoted. Hungary has also taken legal action.
Despite the V4's determination to stand by its common positions, Fico said the group would "never go against the EU."
"We will have our original positions, but we will not push it at the price of damaging the EU," he said.