-Britain's prime minister has begun talks with a Northern Ireland-based party Tuesday to see if they can create an alliance to push through the Conservative Party's agenda after a disastrous snap election left her short of a majority in Parliament.
But a deal with the DUP would risk destabilising the political balance in Northern Ireland by increasing the influence of pro-British unionists who have struggled for years with Irish Catholic nationalists who want Northern Ireland to join a united Ireland. May would remain the prime minister of the country while the DUP MPs would play a critical role in helping the Conservatives remain in power.
The proposed deal would see the DUP back the Conservatives in votes on the Budget and on any confidence motion while other matters would be negotiated on an issue-by-issue basis.
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.
Speaking alongside the President at the Elysee Palace, she said: "I confirmed to President Macron that the timetable for the Brexit negotiation remains on course and will begin next week".
Another party source told the Guardian the unionists were seeking to make May's government's policies "more compassionate" across the UK.
The negotiations with the DUP revolve around support from the party on a vote-by-vote basis in parliament, rather than a formal coalition government.
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.
Under the DUP, Northern Ireland remains staunchly to the right of the rest of the United Kingdom.
Even the idea of an alliance is complicated, however.
The Commission has stated that one of its priorities in Brexit talks was the Good Friday Agreement, which brought an end to violence and terrorism in Northern Ireland in the late 1990s.
Sinn Fein has declined to say whether it might challenge a Tory-DUP tie-up in the courts, but Alex Maskey, a Sinn Fein member of the Belfast Assembly, said it could "prove to be reckless".
He said: "Any deal which undercuts in any way the process here or the Good Friday and the other agreements is one which has to be opposed by progressives".
The stakes for May are high as lawmakers return for their first day of business Tuesday.
Ministers have already said that the Queen's Speech may have to be set back from its scheduled date of next Monday June 19, because of the ongoing negotiations.
Labour leader Jeremy Corbyn countered with a bit of previously unforeseen swagger, wearing a huge red rose - his party's symbol - in his lapel as he sparred with May.