The DUP leader, Arlene Foster, said she hoped to shake hands "sooner rather than later".
The DUP leader 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".
"We will go in to speak with Sinn Fein again on Monday morning to try and get that set up as quickly as possible, because devolution works and works for everybody in Northern Ireland".
Though Foster supported Brexit, she also might demand that May pursue a cushioned exit from the European Union, given her party's wish that a soft border remain between Northern Ireland and Ireland, an European Union member.
"We know each other and we understand each other".
While the DUP are deeply euroskeptic, they have balked at some of the practical implications of a so-call hard Brexit-including a potential loss of a "frictionless border" with the Republic of Ireland-and talks will touch on efforts to minimize the potential damage to Northern Ireland. The DUP has said there is no deadline for the conclusion of these talks.
The former Conservative prime minister, Sir John Major has cited his concerns over the Northern Ireland peace process urging Theresa May to walk away from doing any deal with the DUP.
Talks between the Conservatives and the DUP are reported to be progressing well, although it is likely to be early next week before an agreement is finalised.
Gerry Adams led a delegation to Government Buildings including Ms O'Neill, Sinn Fein deputy leader Mary Lou McDonald and newly-elected Fermanagh South Tyrone MP Michelle Gildernew.
Scottish Conservative leader Ruth Davidson said the government should put economic growth at the heart of its Brexit strategy, comments placing her in the camp of those advocating a closer trade relationship with the European Union, or "soft" Brexit.
He told the Press Association the Government could not act as a "neutral facilitator" in Northern Ireland, as the Good Friday Agreement envisages, if it was dependent on one of the Northern Irish parties for its majority in the House of Commons.
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.
Mr Brokenshire insisted the Government would honour its commitments in the Good Friday Agreement and warned that time was running out if power-sharing was to be restored and a return to direct rule from Westminster avoided. However, "a little side bargain to keep Theresa May in power, a temporary little arrangement, won't have any integrity".
May will host Arlene Foster in Downing Street to discuss terms of the DUP's backing for her minority government, reports the BBC.
But any deal between the DUP and the Tories could worsen tension in Northern Ireland and open the door to a referendum on unity with Ireland.
UK's Hammond says economy should be priority in Brexit talks
European Union leaders have been calling for talks to begin - on Tuesday Mr Barnier said: " I can't negotiate with myself ". Duke of York said he has "no idea" how Brexit will play out but said there may be "fresh grass" for British businesses.
Rickie Fowler leads US Open on day of highs and lows
But they also know that Erin Hills could strike back if the wind kicks up and the course dries out a little or if the U.S. Scottie Scheffler, who just finished his junior year at Texas, had a 69 and amateur Cameron Champ had a 70 in his U.S.