"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'm convinced a deal is necessary, I'm still not sure we'll get one", he said.
Earlier, Ireland's deputy prime minister Simon Coveney confirmed the EU's chief negotiator Michel Barnier was prepared to extend the 21-month transition period after Britain leaves the bloc to allow time to negotiate an alternative arrangement.
European Union diplomats said May appeared to show a greater understanding of the EU's concerns, including Ireland's requirement for a "backstop" insurance clause that would mean there would beno hard border between Northern Ireland and the Republic of Ireland.
It might be hard for May to convince her Cabinet and parliament of an extension of the transition period, however.
British Prime Minister Theresa May said Thursday she is considering a European Union proposal that would keep Britain bound to the bloc's rules for more than two years after it leaves, and idea that angers her pro-Brexit critics in the United Kingdom.
British Prime Minister Theresa May says the remaining differences on Brexit terms between the United Kingdom and the European Union are "few but considerable" - but she says she is confident the two sides will reach an agreement.
"A further idea that has emerged - and it is an idea at this stage - is to create an option to extend the implementation period for a matter of months, and it would only be for a matter of months", she said.
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At the meeting in Brussels, EU leaders decided that not enough progress has been made in negotiations to warrant calling a one-off summit in November for a divorce deal to be signed.
Speaking after hearing Mrs May's address to EU leaders, European Parliament president Antonio Tajani said: "There was a message of goodwill and readiness to reach an agreement, but I didn't perceive anything substantially new in terms of content as I listened to Mrs May".
The possibility of extending the transition period, set to end by December 2020, was also mooted at the Brussels summit.
"That's very much a decision for the British people and the British government", he said.
Juncker and May, however, remain hopeful that the Brexit deadlock could be broken and an agreement reached.
Leaders of other member countries said the real fight was not with the European Union. but within May's fractured Conservative Party.
But she failed to respond to a call by EU President Donald Tusk for "concrete proposals" of her own to move the talks forward.
May spoke a day after Tusk implored her to present new ideas for resolving the tricky problem of how to keep the land border between the Republic of Ireland and the U.K.'s Northern Ireland friction-free once Britain no longer is an European Union member.
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Varadkar added that working out all the details of the deal on the EU-UK relationship could take more than two years.
Addressing leaders of the 27 remaining European Union member states in Brussels, she stressed that significant progress had been made in many areas of the negotiations and urged them to find a "creative" way out of the current dilemma.
The issue of Northern Ireland in these talks is a conundrum that Tusk compared to a "new version of the Gordian knot".
Malmstrom said the Europeans "see this merely as preparations being made by the U.S.to negotiate with them and others".
Both sides agree there must be no hard border that could disrupt businesses and residents on both sides and undermine Northern Ireland's hard-won peace process.
To solve the Irish question, Britain has proposed staying aligned to the EU's customs rules until a wider trade deal can be signed that avoids the need for any frontier checks.
"We want a Brexit that is good for the union in all of its constituent parts, but that also respects the constitutional and economic integrity of Northern Ireland", she added.
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Nine Mega Millions tickets from Tuesday's drawing matched five white balls to win the second prize - $1 million. Believe it or not, this week's combined billion-dollar haul is not the biggest prize in the lottery's history.