Video report by ITV News Europe Editor James Mates
Theresa May will return to Brussels this week hoping to break the deadlock that dramatically stalled Monday's Brexit talks over the question of Northern Ireland's border.
The talks broke down when the PM pulled out of a deal at the last moment after meeting fierce resistance from Unionists to proposals which would align Northern Ireland's regulations with the Republic, Ireland's Prime Minister Leo Varadkar has claimed.
Chancellor Philip Hammond, arriving in Brussels for talks with EU finance ministers on Tuesday, said the negotiations were "very close but not there yet".
"This is a very complex set of negotiations, there are many moving parts ... discussions are going on right now and will go on throughout the day," he said.
Those discussions are likely to be fraught.
Mrs May and European Commission President Jean-Claude Juncker hope to reconvene later this week for more talks ahead of the summit of the European Council on December 14.
The prime minister said she was "confident we will conclude this positively".
"I'm still confident that we can reach sufficient progress before the European Council of December 15," she said.
"This is not a failure, this is the start of the very last round. I'm very confident that we will reach an agreement in the course of this week."
Mr Juncker called the meeting "friendly and constructive".
He went on: "I have to say that she's a tough negotiator, and not an easy one, and she's defending the point of view of Britain with all the energy we know she has, and this is the same on the side of the European Union.
The breakdown came after Mrs May had earlier agreed there will be "no regulatory divergence" between Northern Ireland and the Republic of Ireland after Brexit, a major compromise intended to ensure there is no hard Irish border."
Taoiseach Leo Varadkar said that he was informed that agreement had been reached on the key issue of the Irish border before the dramatic intervention of the DUP.
Regulatory alignment between Northern Ireland and the Republic would likely mean both sides following the same rules governing trade, to ensure that goods can continue to move freely across a "soft" border with no checks.
News of the possible deal angered the Democratic Unionist Party, which said it will not accept any Brexit deal that "separates" Northern Ireland from the rest of the UK.
Video report by ITV News Political Reporter Romilly Weeks
Responding to reports of a draft agreement for there to be "regulatory alignment", Downing Street insisted that the UK's "territorial and economic integrity will be protected".
DUP leader Arlene Foster, whose party is effectively keeping Mrs May in Downing Street in a confidence and supply deal with the minority Tory Government, said Northern Ireland must leave the EU "on the same terms as the rest of the UK".
"We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom," she said.
"The economic and constitutional integrity of the United Kingdom will not be compromised in any way."
Prominent Tory MPs also voiced opposition to the deal. Leading Brexiteer Jacob Rees-Mogg said: "I don't think that can possibly happen. The Government doesn't have a majority for that."
Nicola Sturgeon responded to the reports by saying there was no good reason that Scotland could not do the same with the EU and "effectively stay in the single market".
Leanne Wood, the leader of the Party of Wales, urged the Welsh Government to fight for any special deals afforded to Northern Ireland.
London Mayor Sadiq Khan also tweeted that a similar deal for London would save "tens of thousands of jobs".
Meanwhile, the Police Federation for Northern Ireland warned it does not have the manpower to protect a hard Irish border.
"Numbers are already painfully thin on the ground and, if hundreds were required along a porous border, the situation would inevitably become intolerable and unmanageable," federation chairman Mark Lindsay said.
He warned that planning for a no-deal Brexit needs to begin urgently as it would cost tens of millions of pounds to secure Northern Ireland's porous 300-mile (483km) frontier.
The PM's talks with Mr Juncker and European Council president Donald Tusk could have a crucial bearing on whether she is able to secure the necessary "sufficient progress" at the Brussels summit.
The two sides have appeared to be moving closer on the divorce bill and future citizens' rights but the Irish Government made clear ahead of talks that the border issue - the third area where Brussels is demanding progress - remained unresolved.
Mr Juncker held talks with Taoiseach Leo Vardakar minutes before meeting Mrs May.
The UK government wants the European Council summit of EU leaders on December 14 and 15 to agree that Brexit talks can move on to trade and a transition deal.
Some unionists in Northern Ireland fear that regulatory alignment could lead to the effective drawing of a new border in the Irish Sea between the province and the rest of the UK, if the Westminster Government decides it wants to diverge from EU rules.
DUP Brexit spokesman Sammy Wilson warned Mrs May not to proceed with regulatory alignment.
He said: "I think that this is emanating from the Irish Government, obviously, trying to push the UK Government into a corner in the negotiations.
"It is not well thought through. I don't think, given its promises, the British Government could concede on this."