Ms May is now engaged in talks with the Democratic Unionist Party over a so-called "confidence and supply" arrangement that give the Conservatives a majority in the House of Commons and the ability to form a government.
In exchange for certain concessions which are still being decided, the DUP will agree to vote with the Tories on cornerstone bills such as the Budget and the Queen's Speech, the document which outlines a Government's policy priorities for the year.
With ten seats, a natural right-of-centre political persuasion and a pro-Brexit outlook, Northern Ireland's Democratic Unionist Party became kingmakers for the Tories.
London's neutrality is key to the delicate balance of power in Northern Ireland, which was once plagued by violence over Britain's control of the province.
O'Neill told reporters outside the Dail they are also committed to getting a government up and running in the North.
"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", former Conservative prime minister John Major told BBC radio.
Mrs May has dismissed calls to resign following the dismal election result after calling a vote three years early, in the hope of bolstering her slim majority ahead of Brexit talks starting this month.
"My preoccupation is that time is passing - it's passing quicker than anyone believes".
The European Parliament's Brexit negotiator, Guy Verhofstadt, also expressed his frustration.
Ms Graham addressed the issue of Brexit, and said the EU, London, Scotland and Wales all had a plan for Brexit.
France's 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. "We will continue with those negotiations throughout the weekend and into next week".
The EU meanwhile unveiled plans to give itself new powers over London's banking business after Brexit, in what could be a blow to the city's supremacy as a global financial hub.
The Times newspaper said finance minister Philip Hammond would push May not to leave the customs union - an arrangement which guarantees tariff-free trade within the bloc but prohibits members from striking third-party trade deals.
A working dinner between the British premier and her Europhile counterpart - a Brussels favourite - will be followed by their attendance at a friendly football match between England and France.