After the talks in Dublin, Irish foreign minister Simon Coveney said a deal in November might still be possible, but further progress is required and "we aren't quite there yet", the Sun reports. It is Belfast after "breccia" will continue to obey the Customs Union of the EU and numerous rules of the single market.
They say May should get on with delivering Brexit.
British housing minister James Brokenshire said on Sunday there was still an issue around Northern Ireland in the Brexit talks, referring to a yet-to-be finalised "backstop" arrangement which would prevent a hard border between Northern Ireland and European Union member Ireland if no better solution can be found.
But he made clear he would not accept an arrangement which gave the United Kingdom unilateral powers to ditch the customs union without the agreement of Brussels.
The EU originally proposed keeping Northern Ireland alone under EU customs rules and other economic regulations in order to avoid disrupting the peace in the province by setting up barriers on the Irish border, the only EU-UK land frontier.
"And I think it's reasonable for us to expect a country like the United Kingdom and a government like the UK government to stand by its commitments".
"So, we think there's a deal to be had if they recognise that the deal is unacceptable to Parliament, I think that opens up a vista of the opportunity of the real negotiations".
UK Brexit Secretary Dominic Raab.
The poll showed sentiment shifting in whole cities, including Birmingham, Luton, Nottingham, Slough and Southampton, all now solidly "remain" having originally voted leave.
Mr Gauke told a Channel 4 debate show: "If we leave on no-deal terms there's no good shying away, it will be very bad for us economically".
This was to be a "backstop" insurance clause in the Brexit treaty, to be triggered if, by the end of a status-quo transition period, no better way had been agreed to keep the Irish border operating more or less as it does now.
However, the nation appears to be sharply divided over what Theresa May should do next if she is unable to hammer out a Brexit deal by 29 March 2019, the date when Britain is due to exit the bloc.
"Looks like we're heading for no deal", Jeffrey Donaldson, one of 10 Democratic Unionist Party (DUP) lawmakers whose support May now needs to get any deal passed in the British parliament, said on Twitter.
Remain would win a new Brexit poll by 54%-46%, according to analysis of one of the largest surveys carried out on the issue.
She is expected to warn her most stridently anti-EU MPs that they will be held responsible for the consequences of a no-deal Brexit if they do not vote for her deal.