The EU proposed a "backstop" solution to keep the Northern Ireland in the bloc's customs union and parts of the single market that underpins the 1998 peace agreement there - an option British prime minister Theresa May has rejected so far arguing it would create a border within the United Kingdom, in the North Sea between Northern Ireland and the mainland UK. It must be one of the worrying questions for Downing Street.
The de-facto Deputy Prime Minister was pressed on the DUP's support when he appeared on ITV's Peston on Wednesday night.
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".
London hopes to resolve the issue with a future trade deal, but agrees there should be a "backstop" arrangement to avoid physical frontier checks until that deal is done.
May has ruled out any agreement which could see customs or regulatory barriers being erected between Northern Ireland and mainland Britain.
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.
However, Mrs Foster said the EU's proposals would place "an effective one-way turnstile" between Northern Ireland and the rest of the UK.
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On Wednesday DUP Brexit spokesman Sammy Wilson warned the Prime Minister that his party would pull support and collapse her government if Northern Ireland was left "languishing in the stifling embrace of the EU" after Brexit.
By withdrawing its support, the DUP could make it hard for May to pass legislation through parliament, including the budget which will be voted on later this month.
Mrs Mordaunt refused to say she backs the Prime Minister's Chequers deal during an interview earlier this week with The Daily Telegraph.
The backstop plan being proposed by Mrs May would involve the whole of the United Kingdom remaining in a customs union with the EU while negotiations over a free trade deal take place, which Brexiteers fear could take years. "Such a customs union would eliminate an important part of customs checks".
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.
Eurosceptic Conservative MPs who want a clean break with the European Union have expressed alarm at reports that, as it races to get a deal, the government may agree this alignment would last indefinitely.
Barnier earlier in the day briefed European Union commissioners on the negotiations.