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". "They are ongoing, it hasn't broken up for the day".
The two leaders will hold their bilateral dialogue as they attend a France vs England worldwide friendly football match.
Referring to claims from Sinn Féin that a deal with the Conservatives could represent a breach of the Belfast Agreement, she said she would advise people to read the agreement.
"I'm the person who got us into this mess and I'm the one who will get us out of it", she said.
They are hoping to ramp up the pressure on the region's largest political party to change its stance at a time when its policies on social issues are under renewed scrutiny due to the likely parliamentary deal with the minority Tory government.
"We remain fully committed to making the institutions work", she said.
Understandably, northern nationalists are extremely concerned that, with their new influence, the DUP will seek to scrap numerous concessions nationalists won in the Good Friday agreement.
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Turkish Foreign Minister Mevlut Cavusoglu, on a Gulf tour trying to help broker an end to the crisis, defended the deal.
The government meanwhile said the state opening of the Parliament - when May's government presents its legislation programme - will take place on 21 June, two days later than planned.
Jeremy Corbyn, leader of Britain's main opposition, said his Labour Party would not support May's Queen's speech in the lower house of parliament to try to force her out of power through a vote of no-confidence.
The Conservative source said: "We're confident of getting an agreement, we're confident that the Queen's speech will be passed". "They can't have it both ways, it has to be dealt with sensibly", she said.
The tragic fire in a London tower block looks set to delay an agreement between Theresa May and the DUP about forming a stable government.
It is thought Mrs Foster, despite being a Brexit supporter, could seek assurances from Mrs May that she will pursue a softer exit from the European Union, given Northern Ireland's 56% Remain vote and the DUP's desire not to see a return to a hard border with Ireland.
The start of parliament has been delayed since last Thursday's election, a gamble May took to strengthen her hand in talks to leave the European Union but which has left her scrambling for a deal with the eurosceptic DUP to keep her in power.
Despite the uncertainty over her ability to govern, May had confirmed that Brexit negotiations - expected to be the most complex global talks Britain has held for decades - would begin as planned next week.
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.
It's a unity of goal, having voted to leave the European Union, that their government gets on with that and makes a success of it."During the election campaign, May cast herself as the only leader competent to navigate the tortuous Brexit negotiations that will shape the future of the United Kingdom and its $2.5 trillion economy".