It comes as DUP leader Arlene Foster and the party's Westminster leader Nigel Dodds have been having ongoing discussions with Theresa May to strike a deal for them to work a minority government with the Conservatives.
"There is a unity of goal among people in the United Kingdom", May said, following a meeting with French President Emmanuel Macron in Paris.
As Mrs May attempts to cobble together a majority, the EU's Michel Barnier said he would hold talks with British envoy Oliver Robbins on Tuesday to organise the negotiations.
At least six people died and dozens were injured when a massive fire tore through a 24-story London apartment block overnight, a fresh blow to the capital less than two weeks after the London Bridge terror attack.
On a visit to Paris on Tuesday evening where she met with French President Emmanuel Macron, May described the talks with the DUP as "productive".
Brexit minister David Davis has insisted the approach to the European Union divorce had not changed, but at the meeting with lawmakers on Monday, May recognized that a broader consensus needed to be built for Brexit and made clear she would listen to all wings of the party on the issue.
- Britain is welcome to change its mind and stay in the European Union, but it should not expect to keep getting its EU budget rebates or complex opt-outs from EU rules, the European Parliament's Brexit coordinator said on Wednesday, June 14, Reuters reports.
Mrs Foster said: "We've had some very good discussions today and those discussions are continuing this afternoon". A Number 10 spokesman said: "We welcome this commitment, which can provide the stability and certainty the whole country requires as we embark on Brexit and beyond".
"It's passing quicker than anyone believes".
Putin warns new sanctions will 'complicate' ties between Russian Federation and US
The bill, which passed 98-2, targets Russian individuals accused of corruption and key sectors of the Russian economy. During that process, the sanctions will remain in place unless the Congress votes in agreement with the President.
"We really need to do everything we conceivably can to make sure that that doesn't happen, and that does require an impartial United Kingdom government".
The EU will keep the door open for Britain to return, but only on worse terms than it now has, European Parliament Brexit negotiator Guy Verhofstadt said Wednesday.
Ms May faces a hard balancing act: Divisions over Europe helped sink the premierships of Margaret Thatcher, John Major and David Cameron, and many of her politicians and party membership support a sharp break with the EU.
The Tories need the support of the ten DUP MPs after losing their majority in last week's election disaster.
They would not form a coalition.
Concerns have been raised, including by Sir John Major, that the Government can no longer be the necessary "honest broker" in talks in Northern Ireland, if it is formerly allied with the DUP.
The nationalist Sinn Fein and SDLP and the cross-community Alliance have all made clear Northern Ireland Secretary James Brokenshire can not chair the ongoing process to restore power-sharing at Stormont due to the perceived conflict of interest.
Some observers have maintained that a deal with the DUP risks destabilising Northern Ireland by increasing the influence of pro-British unionists.
The Irish republican Sinn Fein party - which won seven seats in the election although their MPs traditionally do not take up their seats in protest - is also wary of the alliance.