The Conservative leader lost her parliamentary majority in last week's election and is now desperately seeking the backing of the 10 MPs from Northern Ireland's ultra-conservative Democratic Unionist Party (DUP).
"What we are doing in relation to the productive talks that we are holding with the Democratic Unionist Party is ensuring that it is possible to, with their support, give the stability to the UK Government that I think is necessary at this time".
It was reported that diary commitments meant finalisation of any deal could be delayed until next week.
Ministers have indicated that the Queen's speech may have to be set back from its scheduled date of next Monday 19 June because of the ongoing negotiations.
"I will be making it very clear that any deal between the Tories and the DUP can not be allowed to undermine the Good Friday and subsequent agreements", Ms O'Neill said in a statement.
Even without the impact of the London fire, the talks were said to be making "slower progress".
"Discussions are going well with the government and we hope soon to be able to bring this work to a successful conclusion", the DUP's Arlene Foster said after the meeting.
"I will have a problem with it if money which is being given to Northern Ireland for purposes that are devolved to Wales and Scotland", he told BBC Radio Wales' Good Evening Wales programme.
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.
"It's going to be hard".
Nevertheless, it illustrated the challenge Prime Minister May will face - finding a position that satisfies both pro-European and eurosceptic factions of her party - if she wants to remain in power.
In an interview with the Financial Times Mr Barnier said Britain risks crashing out of the European Union in March 2019 without a deal on future relations. "Scotland voted against Brexit".
"I hope the people of the United Kingdom will now have the sort of honest detailed, sector by sector, debate on what Brexit might mean, a debate that they so markedly failed to have during the general election campaign".
While the DUP continue to hammer out the details of the arrangement with the Tories, the other four main parties at Stormont confirmed they would be meeting Mrs May on Thursday.
Asked about Schaeuble's comments, 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.
"But, let us be clear and organised and once the negotiations have started we should be well aware that it'll be more hard to move backwards", he noted.
Mrs May stressed Brexit would happen and the timetable remains on course.
When the Prime Minister sat down for dinner with Emmanuel Macron in Paris Tuesday evening, the new French President - who has set out his stall as an European Union reformer - likely reminded her that the clock is ticking.
"The danger is that however much any government tries they will not be seen to be impartial if they are locked into a parliamentary deal at Westminster with one of the Northern Ireland parties, and you never know in what unpredictable way events will turn out, and we can not know if that impartiality is going to be crucial at some stage in the future".
Mats Persson, a former European Union adviser to David Cameron, told CNN: "What they [Macron and other European Union leaders] want is clarity".