Mr Barnier added: “As far as the freedom of movement is concerned, I can only repeat that the four freedoms are linked together and are not negotiable.”
However, a return to freedom of movement risks sparking public fury amid the Government’s pledge to slash migration.
Re-opening the borders to the bloc would inevitably increase the numbers of migrants coming to Britain, a senior Whitehall source said, unless the Government found another way to reduce the flow.
Leaked documents have suggested the EU could stop Mr Burnham from deporting migrant criminals as part of Labour’s reset deal.
The bloc’s demands have emerged within a draft statement prepared ahead of this month’s crunch summit.
The document, seen by Sky News, shows Brussels will demand Labour “commits not to proceed with… removals and deportations” of migrant criminals in Britain from member state countries.
The push comes after the Home Office was given the green light to deport migrant criminals once they are sentenced earlier this year.
The contents of the draft statement have yet to be signed off by the EU’s member states.
GB News has approached No10 for comment.