"The answer is yes", Barnier said when asked on France Inter radio if the Irish border issue could cause the negotiations to collapse. "I believe we need a deal".
"I think we are both of the understanding that this is an insurance policy and obviously we want to address all of these issues in a future relationship", she said.
But while the mood at a two-day summit in Brussels was more upbeat, there was little movement from either side on how to resolve the border issue, with European Union officials and diplomats saying May had offered nothing new to unlock the talks.
The summit also was meant to lay the groundwork for an additional planning meeting next month for the U.K.'s exit - though, as the BBC writes, it's possible that European Union leaders will formally say they are going to use the November gathering to prepare for that no-deal scenario.
He said he was open to the idea of extending the transition period between the formal Brexit date in March and the date when the UK's new relationship with Europe begins in earnest.
Whether it be staying in the customs union to avoid creating a hard border between Northern Ireland and the Republic of Ireland or organising a comprehensive border around Ireland, May's proposals are being met with strong domestic opposition. Under the current proposal, the United Kingdom would effectively be a non-member: it would remain part of the single market and customs union, but would no longer be represented in EU institutions.
Thomson, who since his election to Westminster previous year has become one of the most vocal and high-profile Tory backbench Eurosceptics, took to Twitter to blast the Prime Minister's plan: "We would be obliged to comply with European Union regulations whilst having no say over them", he wrote.
They concluded that the progress in the EU-UK Brexit talks was not sufficient in spite of multiple negotiations on the matter.
As EU leaders gathered yet again in Brussels yesterday, they conceded that hopes of a deal are again being pushed back, this time perhaps until December.
The main point of contention is the Irish border.
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Theresa May has told business leaders she knows time is running out to reach a Brexit deal but she is confident it can be done, the head of the Institute of Directors has said.
The problem is that a "deal sold in this country as a contingent backstop would be treated by Eurocrats as a definitive settlement", says pro-Brexit MEP Daniel Hannan in the Daily Telegraph.
And one person familiar with the discussions said May's Tory party would find it hard to be fighting the next general election - due in 2022 - while the country is still inside the single market and customs union.
Extending the transition period could mean that if a future partnership is not ready, a backstop, which so far has been unpalatable to the British side, would not have to be triggered.
The Democratic Unionists, however, have threatened "consequences" if the British government does anything which could undermine the constitutional integrity of the UK. We need to know what the other side wants - finally, ' said Lithuanian President Dalia Grybauskaite.
She is hemmed in by pro-Brexit members of her Conservative Party, who oppose any more compromises with the bloc, and by her parliamentary allies in Northern Ireland's Democratic Unionist Party, who insist a solution can't include customs checks between Northern Ireland and the rest of the United Kingdom.
For all her "wittering" at the dispatch box about backstops, she "can get nothing worth having through parliament or the EU".
The Tanaiste said concerns about the potential to destabilise the island's "very precious peace" were real, and reflected the fears of people living on the border.
With divorce talks stuck, the bloc has suggested extending that period, to give more time to strike a trade deal that ensures a frictionless border between Northern Ireland and Ireland.
Many EU leaders "see the delays as a negotiating tactic by London to try to force concessions", Nelson reported.