British and European Union negotiators making headway on the Irish border hope for a Brexit deal breakthrough on Monday, diplomats said, though the British prime minister's Northern Ireland ally has stoked uncertainty by warning it could vote against her.
May gathered selected members of her cabinet late Thursday to update them on the Brexit talks, and the Financial Times reported that she told them that a deal was drawing closer.
Dutch prime minister Mark Rutte said on Wednesday evening he is "cautiously optimistic" that progress will be made next week on the negotiations for Britain to leave the EU.
Both sides have made positive signals in recent days though also indicated they were still short of a deal, which would define the divorce terms and include an European Union declaration stating that it would seek the closest possible ties with Britain after Brexit ranging from trade to security to research.
"They [the EU] said they think this is an unbelievable deal you are getting, something nobody else in the world has even been offered", said Seamus Leheny, the director of policy at the Freight Transport Association of Northern Ireland, who was at the meeting.
Checks of animals and animal-derived products would still have to take place between Northern Ireland and the rest of the United Kingdom and cover all of that trade.
Barnier told small business leaders that the talks were "continuing intensively this week, day and night, with the aim. of having a deal within reach, if we follow through to the end of this negotiations, on October 17". "There is no breakthrough - yet", Margaritis Schinas, chief spokesman for the EU's executive European Commission, told a news briefing on Thursday.
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He added: "If they (the government) decide to cave in to the unreasonable and unnecessary demands which are being promoted by Brussels, then we will have to consider whether or not they have kept their side of the bargain". Another summit in November is planned to close the deal.
While the United Kingdom will effectively stay in the single market and customs union during the transition period up to December 2020, Brexiteers fear the new proposal would keep the United Kingdom in the customs union for years thereafter until a full trade deal could be agreed with the EU27.
The OBR, an independent non-departmental public body, said it was "next to impossible" to predict the consequences of a no-deal Brexit.
The DUP agreed a "confidence and supply" deal to support Mrs May after last year's general election and losing the party's support in the Commons would mean possible defeat on the budget at the end of October and a no-confidence vote.
The DUP has threatened to vote against the government's budget if it goes ahead with the plan.
The EU suggests Northern Ireland stay aligned with its customs union and single market during that period, and there are signs the British government could accept elements of this plan.