The EU's most senior official called for a new approach to the negotiations after extraordinary Sunday afternoon talks between Michel Barnier and Dominic Raab produced a new impasse.
Negotiations are now in a state of deadlock over what form a "backstop" should take to prevent a hard border on the island of Ireland if a trade deal is not ready by the time the transition period expires at the end of December 2020.
Less than six months before Britain is due to leave, the impasse has increased the possibility of a "no deal" Brexit that could potentially disrupt trade, delay movement of goods and starve the world's fifth largest economy of investment.
The EU has stuck to its position - a backstop to prevent a return to a hard border between Northern Ireland and Ireland can not be time-limited and can not include the whole of the United Kingdom remaining in the customs union.
Claiming the backstop plan was merely an "insurance policy" in case a future trade agreement was not ready to be implemented, she said: "We expect that to be no later than December 2021 but we will be working to ensure that that point comes as early as possible". May's Chequers plan for a softer version of Brexit, while 63 Conservative MPs signed a letter accusing the government of misrepresenting the economic impact of a no-deal scenario.
Barnier met his British opposite number Dominic Raab in Brussels on Sunday, but they failed to agree a draft Brexit divorce arrangement before European Union leaders arrive in the city on Wednesday.
Tory MP Simon Clarke, who earlier this year submitted and then withdrew a letter of no confidence in the Prime Minister, said May had "failed to reassure" MPs the United Kingdom would be able to leave the backstop arrangement when she claimed.
He said: 'The decisions we take in the next seven weeks will shape the relationship that Britain has with continental Europe for the next 50 years.
Speaking on the eve of the summit in Luxembourg, Barnier said many terms of the divorce have been agreed upon already, such as how much money Britain owes the EU.
Leaders will decide at dinner after May has left them on Wednesday whether to firm up a tentative plan to hold a special Brexit summit in mid-November. She says that is unacceptable because it would impose new controls between Northern Ireland and the rest of the UK.
Talks on Sunday between British Brexit minister Dominic Raab and the European Union negotiator Michel Barnier ended without breakthrough on the Irish border issue. Things have gotten so bad that May had to convene her Cabinet to get their approval for her negotiating strategy, because that went so well last time. The thinking in Brussels is that she also needs to show politically that she did not agree to the EU's demand, but fought back.
Since the Brexit discussions began over 18 months ago, the October summit was earmarked as the most likely date for an agreement given the need for parliamentary approvals before Britain officially departs in March.
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