May announced Friday she would seek to form a minority government with the help of a small Northern Irish party, the far-right Democratic Unionist Party (DUP).
The Conservatives won 318 House of Commons seats in Thursday's election, eight short of an outright majority.
Northern Ireland's Democratic Unionists were in talks Friday to become kingmakers by lending British Prime Minister Theresa May enough backing for a parliamentary majority - support that throws the spotlight on the party's deeply right-wing outlook.
"Theresa May is a dead woman walking".
Speaking after a 20-minute audience with the Queen, Mrs May said she meant to form a new administration to take Britain into the Brexit negotiations beginning on June 19.
"I think her position is, in the long term, untenable", another Conservative lawmaker, Anna Soubry, tells Sky News.
Downing Street said it hopes to finalize the deal next week, after Parliament resumes sitting.
They added that both parties "confirmed their joint commitment to restoring a Northern Ireland Executive as soon as possible and agreed that both countries would continue to engage closely to bring about political stability in Northern Ireland". "They know exactly what they want, but are never going to get too close to us", said a source.
The outcome risks upsetting the political balance in Northern Ireland by aligning London more closely with the pro-British side in the divided province, and increases the chance that Britain will fall out of the European Union in 2019 without a deal.
"You've got to convince them of your credibility and that you can move from protesting about the government to being in government", he said.
"This is not the time for sharks to be circling".
Afghanistan To Host International Peace Conference
He said that up to 11,000 foreign militants had allegedly arrived in Afghanistan in the last two years to fight for Daesh. Afghan authorities have said Pakistan was involved in the attack, charges denied by Islamabad.
Ms O'Neill said: "Sinn Fein's focus remains on entering talks to re-establish an Executive which delivers for all on the basis of equality, respect and integrity, and this requires the full implementation of agreements on rights and legacy".
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In another sign of the dangers facing Mrs May, Sunday papers reported that Boris Johnson was either being encouraged to make a leadership bid in an effort to oust her, or actually preparing one - a claim dismissed as "tripe" by the Foreign Secretary.
But a dismal campaign has left the Prime Minister fending off a mutiny in her own party.
The results have seen even normally loyal MPs criticise their party leader. "So we have 15, or 16 more months".
Under pressure from Conservative cabinet ministers, May accepted the resignation of her two top aides, her co-chiefs of staff Nick Timothy and Fiona Hill, on Saturday.
"I'm sorry for all those colleagues who lost their seats and didn't deserve to lose and of course I'll reflect on what happened", May, her voice quavering, said in a pooled interview to broadcasters.
May has insisted she will not step aside, and will instead form a new government that will lead the country through the treacherous currents of the Brexit talks to come.
He told ITV's Peston on Sunday: "Well, not necessarily the case".
Labour leader Jeremy Corbyn, riding a wave of acclaim for his party's unexpectedly strong showing, called on May to resign.
Mrs May needs support in Parliament because the Queen's Speech setting out the Government's programme is due on June 19, with a crucial vote on it expected after a few days' debate.
"It is quite possible there will be an election later this year or early next year and that might be a good thing because we can not go on with a period of great instability", Corbyn told the BBC.
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