"They are close to despair at the state of this negotiation because there is a fear that both the Government and the European Union are trying to run out the clock, that they are trying to leave this so late that they can credibly say there is no alternative but a no-deal Brexit, and most people agree that would be chaos".
But with Brexit talks at an impasse, the bloc has suggested extending that period, to give more time to strike a trade deal that ensures the border between Northern Ireland and the Republic of Ireland remains friction-free - the main sticking point to a Brexit deal.
Chancellor Angela Merkel's spokesman said that members also discussed cybersecurity, migration, and the continued development of the Eurozone.
The so-called "moment of truth" in the Brexit process - the October Brussels summit - has been and gone.
Sinn Féin's Michelle O'Neill said the prime minister was "ignoring majority voices" and SDLP leader Colum Eastwood said it was "disgraceful". "The key element is a British political compromise".
'If you have a negotiator who has no strong mandate, it's very hard to negotiate'.
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U.K. Prime Minister Theresa May said she is weighing a plan that would keep the U.K. bound to European rules for longer, in an attempt to break the deadlock in talks.
EU Council President Donald Tusk said the summit also left him more optimistic than he'd been after a bad-tempered meeting last month and a negotiating crisis on Sunday.
The agreements on the final Brexit deal should be reached by November 16 - the deadline to hold a special summit with the European Union and sign a draft agreement.
Britain needed "to decide finally what they want and to rally behind the prime minister all together, not split", she said, adding: "Today we do not know what they want".
An further year could help the United Kingdom and the EU negotiate a future trade deal, and could ease the burden for May on agreeing to a so-called backstop on the Irish border issue that would keep Northern Ireland in the customs union and the single market for goods.
A series of recent European Commission proposals on migrant returns, a European asylum agency, and the European border and coast guard were flagged by European leaders as issues the European Parliament and European Council should examine with priority.
Unlike the special protocol on Gibraltar, which will be annexed to the UK/EU Withdrawal Agreement, the memorandums are not an essential part of the Brexit deal. It is a good idea.
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"I'm clear that it is possible to do that and that is what we are working for".
The backstop is a proposed mechanism to keep the United Kingdom in the EU's Customs Union after Brexit - essentially precluding the country from taking back control of its global trade policy - in order to prevent a hard economic border between the British province of Northern Ireland and EU member the Republic of Ireland.
On the Remain side, The People's Vote campaign, which is demanding a second referendum, said extending the transition by a year would cost the United Kingdom at least an additional £8.2billion on top of the already-agreed £39billion "divorce bill". Many of them suspect the British leader is pursuing a strategy of delay created to push negotiations to the brink, at which point Europe would yield and the hardcore Brexiteers in the British Parliament would be hard-pressed not to accept May's version of withdrawal, even if they despise it as a capitulation.
"We have shown we can do hard deals together constructively", the PM said.
In a press conference at the end of the summit, Mrs May said good progress was being made on the UK's withdrawal agreement, but acknowledged that "a few, but considerable, outstanding issues" remained to be resolved.
Merkel said: "Where there's a will, there should be a way". He acknowledged the feeling "may be a more emotional impression than a rational one".
At present the two sides are proposing that Britain remain inside the European Union single market and bound by its rules from the time it leaves the bloc in March until December 2020, to give time for new trade relations to be set up.
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