DUP leader Arlene Foster has indicated a deal to support the Conservatives' minority government is close to being finalised.
Speaking after hours of talks between the two parties, the former First Minister said Brexit, counter-terrorism and "doing what's right" for the Northern Ireland economy were among the key issues being thrashed out.
Discussions in Westminster are continuing without Theresa May after she left for Paris for a pre-arranged meeting with newly elected president Emmanuel Macron.
Mrs Foster said: "We've had some very good discussions today and those discussions are continuing this afternoon.
"I hope that we can reach a conclusion sooner than later."
Mrs Foster arrived in Downing Street with Nigel Dodds at lunchtime for negotiations with the Prime Minister but they decamped to Parliament to allow Mrs May and the DUP's deputy leader to speak in the Commons.
Chief Whip Gavin Williamson, who flew to Belfast for talks with the Unionists at the weekend, is continuing to lead for the Conservatives.
Mrs Foster said: "There's been a lot of commentary around the issues that we are talking about and it won't surprise anyone that we are talking about matters that pertain, of course, to the nation generally.
"Bringing stability to the UK 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."
A Downing Street source said the talks had been "constructive" but refused to put a timescale on when they would conclude.
"It'll be done when it's done," they added. "Talks are going well."
Mrs May left for the Commons for the election of the Speaker without answering reporters' questions about how negotiations were progressing.
The Prime Minister did not mention the ongoing deliberations as she addressed MPs but called on Parliament to "come together in a spirit of national unity" to deal with the challenges facing the country.
Labour leader Jeremy Corbyn turned Mrs May's election slogans against her, claiming a link-up between the Tories and DUP would be a "coalition of chaos".
He said: "I'm sure we all look forward to welcoming the Queen's Speech just as soon as the coalition of chaos has been negotiated.
"If that's not possible, the Labour Party stands ready to offer strong and stable leadership in the national interest."
A failure to gain support from the Northern Irish party would risk the Queen's Speech being voted down next week, and Mr Corbyn has said Labour will be pushing hard for that outcome.
Sinn Fein has warned such a move undermines powersharing talks in Northern Ireland and the party's seven MPs have flown to London where they will hold a briefing with reporters.
Conservative former Prime Minister Sir John Major, who was crucial in bringing peace to Northern Ireland, has raised concerns about the impact of a Tory deal with the DUP.
He told BBC Radio 4's The World at One programme: "The last thing anybody wishes to see is one or other of the communities so aggrieved that the hardmen, who are still there lurking in the corners of the communities, decide that they wish to return to some form of violence."
The DUP is taking a "significant risk" in entering a possible deal with the Conservative government, Tory peer Lord Trimble has suggested.
The peer, who was first minister of Northern Ireland from June 1998 until October 2002, said the party was "putting themselves in a position where they may have to take responsibility for unpopular actions".
However Lord Trimble, leader of the Ulster Unionist Party from September 1995 until May 2005 and one of the architects of the Good Friday Agreement, said the Government was also "taking a risk" .
Asked if talks put the Agreement at risk, he said "no", adding: "I think there's a fair amount of scaremongering going on."
Theresa May and Democratic Unionist Party (DUP) leader Arlene Foster were to hold critical talks later on a deal to prop up a Tory minority administration after the Government admitted the Queen's Speech could be delayed.
The Prime Minister will be desperate to get agreement from the DUP to back her legislative programme in the House of Commons or risk her Government falling.
Lord Trimble told BBC Radio 4's Today programme: "There's no connection between the Agreement and the European Union, and Brexit and all the rest of it, people are just trying to grab this and argue as a stick to beat the Government with and I think it's really quite silly."
He added: "The Government has actually acted in good faith and I think again this is just being, trying, looking for something with which to attack the Government and what they should do is to concentrate on the realities of the situation."
Asked about the DUP's demands in any discussions of a deal with the Conservatives, and the Government's stance as an honest broker, he said: "That is the danger for the DUP, because the DUP by doing this are putting themselves in a position where they may have to take responsibility for unpopular actions, where they may suppose themselves to the sort of attack that you're making at the moment, so they are taking a significant risk in going in.
"Equally the Government is taking a risk by coming into a close relationship with them for obvious reasons."
Asked if any deal should go ahead, he said: "They are perfectly entitled to do it, this is not in any way different to what (James) Callaghan did in his arrangements with the liberals way back in the 70s, it's not any different to what Nick Clegg did."
Commenting on whether the talks might result in the restoring of power sharing institutions due to the threat of the alternative of direct power controlled from Westminster by a Government involved with the DUP, he said: "Well that might be a positive consequence, I think we should be grown up about this."