"We are talking in a matter of months, we are not talking in a matter of years", he said.
Mr Barnier said that a Brexit accord was "possible but difficult", adding he was still not certain if a deal could be reached.
Tusk also said the leaders expressed their resolve to stop the flow of illegal migration.
But for now the absence of a breakthrough will be less important to the prime minister than the absence of a breakdown.
However, there was anger among Tory Brexiteers after The Daily Telegraph reported that Chancellor Philip Hammond warned the United Kingdom could still have to pay the European Union up to £36 billion of the £39 billion "divorce bill" to settle its outstanding liabilities, in the event of a no-deal break. "Such a regime should help to protect our citizens, companies and institutions from all kinds of cyber security threats", he said.
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". 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.
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Members of the far-right Proud Boys were involved in a violent incident in New York City on Friday (October 12). Videos posted on YouTube show clashes between the Proud Boys and groups that were protesting McInnes' speech.
Leaders of other member countries said the real fight was not with the European Union. but within May's fractured Conservative Party.
"We are working with European Union to deal with this issue of ensuring that if there is a gap between the end of the implementation period and the point when the future relationship comes in - and we don't expect a gap to exist - we want to ensure there's no hard border between Ireland and Northern Ireland, " May said.
There's significant reluctance from Brussels to drawing a hard border between Ireland and Northern Ireland.
The letter signed by former British Foreign Secretary Boris Johnson, ex-Brexit Secretary David Davis and other pro-Brexit Conservatives warned May not to "engage in a show of resistance and a choreographed argument followed by surrender" to the EU.
She also faces a rebellion from her parliamentary partners, Northern Ireland's Democratic Unionist Party, which has threatened to vote against her government's budget if she concedes to the EU's demands over the backstop.
Its co-chair Richard Tice said: "The original transition was an unnecessary trap created by our weak civil servants who can not be trusted as they don't want us to leave".
Mrs O'Neill said: "In order to preserve her toxic alliance with the DUP, she is trampling over the rights of citizens by acquiescing to that party's refusal to share power (at Stormont) on the basis of equality".
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Addressing fellow European Union leaders in Brussels, the Taoiseach related the story of an IRA bomb attack at a customs post in Newry, Co Down in 1972.
Tusk advised May that "creative" thinking from Britain was required to avoid a hard border on the island of Ireland, the issue that has brought divorce negotiations to a standstill.
The summit in Brussels had always been seen as the "moment of truth" in the two-year Brexit process.
A summit of European Union leaders in Brussels on Wednesday failed to make decisive progress in reaching an agreement.
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.
Labour leader Jeremy Corbyn accused Mrs May of failing "to meet the deadline she herself set for a Brexit deal". Trade negotiations are complex: the more sectors they cover, the greater the number of stakeholders.
The idea of extending the transition would be to give more time to negotiate a deal on future relations and find a formula to defuse the Irish border question.
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The Blast reports that Ari needs time to "work on herself" right now and she needs to be single to do that. Grande reported reportedly pulled out of the event only 24 hours beforehand.