Hopes for a Brexit breakthrough have been kept alive after a suggestion from the Irish deputy premier that a deal could be struck as early as this week.
Boris Johnson is coming under pressure to concede more ground to Brussels, and it has been reported that the UK has dropped a demand that a deal should include a veto for the Stormont Assembly on customs arrangements.
UK and EU officials will continue talks in the Belgian capital on Monday, with the prospects of an agreement in time for Britain to leave with a deal on October 31 in the balance.
Time is rapidly running out if there is to be an agreement to put to EU leaders to sign off on at their two-day summit starting on Thursday.
The EU’s chief negotiator, Michel Barnier, said “technical-level” talks between officials over the weekend had proved “constructive”, while Irish deputy premier Simon Coveney urged caution and said “we’re not there yet” – but added that “a deal is possible”.
The Prime Minister’s spokesman said on Monday that there was still “a lot of work” to be done, but said talks “remain constructive”.
“In terms of the substance of the talks, I’m not going to get into those at all. I think it’s important that people are given space for the talks to progress,” he said.
Downing Street confirmed talks have taken place between the PM and the DUP, with the spokesman saying he would expect those discussions to continue.
Speaking on Monday morning in Luxembourg, Mr Coveney said: “On Brexit the less we say now the better. But we need to give time and space to Michel Barnier and his taskforce and the British negotiating team.
“I think it’s pretty clear what we’re trying to do, but there are pretty detailed technical discussions now and I think we need to give the time and space for that to happen.
“Hopefully we can make progress today on those.”
He added: “The message I would give is that we need to be cautious. This is not an easy job. We’ve spent three years trying to get an agreement between the two sides and have made progress at different times.
“But certainly the last number of months have been difficult. So I think, as my Taoiseach has said, a deal is possible, and it’s possible this month, may even be possible this week. But we’re not there yet.
“And, as Michel Barnier said yesterday, there’s still a lot of work to do, so I hope that we can make more progress today.”
Mr Coveney added: “Our priority is to try to provide certainty to so many people who are awaiting a Brexit outcome and to do that as soon as possible.”
Sinn Fein president Mary Lou McDonald spoke to the Prime Minister about the Stormont veto idea by telephone on Sunday, and told BBC Radio 4’s Today programme: “He assured me, or sought to assure me, that there would be no vetos afforded to anybody in this process. So I can only take him on his word on that matter.”
Mr Johnson has told senior ministers that, while a “pathway” to a deal could still be seen, there was “still a significant amount of work to get there”.
In a Cabinet conference call, he said they still had to be prepared to leave on Halloween without a deal.
And Leader of the House of Commons Jacob Rees-Mogg has said the Government could achieve a no-deal Brexit by using European law.
“Theresa May got an extension not through UK law but through EU law and, until the 1972 European Communities Act is repealed, EU law is superior law in the UK,” he said on BBC Radio 4’s Westminster Hour.
“And the Remainiacs all know that, because they know that it takes two to tango and any extension has to be agreed by the council.”
The sticking point remains the issue of the Northern Ireland backstop intended to guarantee there is no return of a hard border with the Republic.
Mr Barnier was reported to have raised concern about the complexity of a British plan to keep Northern Ireland in the UK customs territory while avoiding the need for border controls.
There were reported to be doubts about the feasibility of the scheme which was said to involve tracking goods as they move through Northern Ireland and then determining the tariff to be paid depending where they end up.
It raised the prospect that negotiations could carry on after this week, with the possibility of an emergency EU summit at the end of the month to finally approve any 11th hour agreement.
However, if Mr Johnson cannot get a deal by the weekend, he will come under intense pressure to seek a further Brexit delay, something he has vowed not to do.
Labour has warned that, if necessary, it will take action through the courts to force the PM to comply with the co-called Benn Act, which requires him to request an extension.
Either way, the stage is set for a major Commons showdown when Mr Johnson returns to Westminster for an emergency Saturday sitting of Parliament, the first in 37 years.
The Number 10 spokesman said Mr Johnson will have to decide by Wednesday whether to propose that Parliament convene on Saturday.
A “sitting motion” will need to be tabled by Wednesday at the latest so it could be voted on by MPs on Thursday.
If Mr Johnson cannot get a deal, he is widely expected to blame MPs for cutting the ground from under him, paving the way for a “people versus Parliament” general election.
If he is able to get an agreement, Government sources have said they will seek to rush through legislation to ratify it in time for the promised Halloween withdrawal date.