She hosted DUP leader Arlene Foster at Downing Street to thrash out the terms of the Northern Ireland party's backing and agreement to vote with the Tories on policies in the House of Commons to get bills through.More news: Premier League fixtures released for 2017-2018 campaign
The 1998 Good Friday Agreement commits the United Kingdom and Irish Governments to demonstrate "rigorous impartiality" in their dealings with the different political traditions in Northern Ireland.
Though on the surface, Thursday's meetings with Northern Irish parties were aimed at breaking the logjam in forming a new cross-party regional government in the province, May needs broader acceptance in the province of a Conservative-DUP deal.
A conclusion on the deal was expected on Wednesday, but the decision may now be delayed to next week, after the DUP said it would be inappropriate to conclude talks and announce a deal on the same day as a fire consumed a tower block in west London, killing 12 people.
But 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.More news: India's May WPI falls to 2.17%
The border between Britain's Northern Ireland and the Republic of Ireland is also a major issue in the talks over Britain's departure from the European Union.
The report cited unnamed sources, and the finance ministry declined to comment.
During the appearance, Ms O'Neill said: 'We made very clear to the prime minister that any deal between herself and the DUP can not undermine the Good Friday agreement.
"I think it's very self-interested, putting party before peace in Northern Ireland - party interest before peace and stability in Northern Ireland - and I do think it's that serious".
May, who ahead of the June referendum supported remaining in the European Union, has promised to start the Brexit talks next week but opponents of a sharp break with the European Union took her woes as a chance to push back against her strategy.More news: Deputy Attorney General Rod Rosenstein: Robert Mueller Won't Be Fired Without Cause
Speaking on BBC Radio 4's World at One, John Major said in regards to the peace process that "People shouldn't regard it as a given".
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