Speaking earlier on Tuesday, DUP leader Arlene Foster told reporters that discussions centered on "bringing stability to the United Kingdom government in and around issues around Brexit, obviously around counter-terrorism, and then doing what's right for Northern Ireland in respect of economic matters".
Prime Minister Theresa May has made a joke at her own expense as Britain's House of Commons got underway - a reflection of her new humility following a disastrous snap election in which she lost her majority. "They are going well".
May will then travel to Paris for a working dinner with French President Emmanuel Macron before the two leaders attend a friendly football match between England and France at the national stadium in Paris. Although it campaigned for Brexit, it is aware that more than 50 percent of Northern Ireland citizens voted to remain in the European Union referendum, June 23, 2016.
Mrs May's botched election gamble has left her so weakened that her Brexit strategy is the subject of debate within her party, with former prime ministers John Major and David Cameron calling on her to soften her Brexit approach. Ms Foster said she hoped a deal could be done "sooner rather than later".
"I think there is a unity of objective among people in the United Kingdom".
It comes as the party has accused the DUP of betraying the interests of Northern Ireland by agreeing to prop up a Conservative minority government.More news: Irish, British PMs to meet to talk Brexit, N.Ireland crisis
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She raised concern that such a deal could put the Northern Ireland peace process at risk as the Government would no longer be an "honest broker" in the Good Friday Agreement.
France's Macron said the EU's door was still open for Britain as long as the negotiations were not finished, but that it would be hard to reverse course.
"I am concerned about the deal. And the danger is that however much any government tries, they will not be seen to be impartial if they are locked into a parliamentary deal at Westminster with one of the Northern Ireland parties".
May held talks with the Democratic Unionist Party (DUP) on Tuesday.
Jeremy Corbyn, leader of the opposition Labour Party, which saw its number of parliamentary seats and share of the vote increase, said there could be another election this year or early in 2018 after last Thursday's vote produced no clear victor.
"The Tory civil war on the European Union which has ripped it apart since the Maastricht rebellions of the early 1990s, and which the referendum was supposed to solve, is now raging again", said Chris Grey, an academic who specialises in Brexit at Royal Holloway in London.