British Prime Minister Theresa May is still hopeful of negotiating a withdrawal deal from the EU.
In an open letter Thursday to May, leading Brexiteers accused the European Union of "bullying" and said the border issue was being used as "a trap" by the bloc. "There is a message of goodwill", said Antonio Tajani, president of the European Parliament. They include workers' movements across the border and security matters.
But she acknowledged that differences remain over the key issue of the "backstop" arrangement to keep the Irish border open in the absence of a broader trade deal.
Her aides say she has had little choice but to countenance an extension of talks and a prolonged transition amid mounting British government alarm that European Union leaders and the bloc's other national governments are close to pulling the plug entirely on Brexit negotiations, and are ready to accept that Britain may crash out of the bloc without any kind deal.
Mr Hunt told Today the transition was not the main issue that needed to be resolved, which remained the so-called Northern Ireland "backstop" meant to ensure there was no return of a "hard border" with the Republic.
"But I still believe that this agreement is possible and our goal is to find the solution and to find the way".
Bettel and European Union commission president Jean-Claude Juncker said they would not oppose it.
If the December summit breaks up without an agreement, there could still be some mini-deals in areas such as aviation to avert a potential catastrophe.
Tom Newton Dunn, political editor of the Sun newspaper, commented that May saying a transition extension would only be for a few months feels like the "I'm only half pregnant" defense. The terms of that, including whether the United Kingdom would have an independent trade policy, could only be agreed after the United Kingdom has left.
The paper also reported her comments during yesterday's post-summit news conference, in which she said: "We all need to find an answer on Ireland and Northern Ireland". "Where there's a will, there should be a way, and normally there is a way".
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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.
European leaders are now discussing another summit in November - or even December before the holidays - to secure a possible accord.
But the two sides disagree on how to resolve the issue.
May's suggestion that she'd be willing to extend the Brexit transition period by up to one year has predictably led to outrage among Conservative MPs and cries of "I told you so" from the opposition.
For many who campaigned to leave the European Union, the idea of continuing to pay into the bloc well beyond Britain leaves in March next year is anathema.
He also described the internal politics of the British Government as "difficult", adding that Prime Minister Theresa May is running a minority government with internal divisions.
He says "we are in a much better mood than after Salzburg".
Seeking to unblock stalled divorce talks with the EU, May said on Thursday that a proposed 21-month transition period after Brexit could be extended by "a matter of months".
Such a border would violate the Good Friday Agreement (GFA) or Belfast Agreement that Northern Ireland's present devolved system of government is based on.
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".
Wednesday morning, Bettel said that "the Brits will be out on 29 March -London has to take decisions now".
May said the United Kingdom was considering extending the transition period by "a matter of months".
It has compounded her problems with a large rebel faction within her own party, however, and other hardcore Brexiters who fear negotiation delays and agreeing to keep Britain tied to the bloc for longer could mean the country, which is deeply divided over Brexit, ultimately never leaves.