Theresa May and Democratic Unionist Party (DUP) leader Arlene Foster are holding key talks as the Tories look to prop up a minority government.
The DUP head has begun negotiations with the prime minister at Downing Street after meeting with her party's fellow Northern Irish MPs in London.
Mrs May is seeking to finalise an agreement with Mrs Foster and her party in a bid to get backing for her legislative programme in Parliament - or risk seeing her government fall.
Any hope for a continuation of Tory rule looks set to depend on a consensus with the Northern Irish party, following last week's General Election result in which the Conservatives dramatically lost their majority.
And the pressure on Mrs May has increased even further after the Government admitted the Queen's Speech could be delayed.
The speech would be at risk of being voted down if the DUP did not reach an agreement with the Tories.
Currently, the Tories and the DUP are considering a "confidence and supply" arrangement which would see the loyalist party back the Government to get its Budget through and on confidence motions.
Talks come after Mrs May told Tory MPs: "I'm the person who got us into this mess and I'm the one who will get us out of it."
But minister Damian Green has confirmed the Queen's Speech setting out the Government's programme could be delayed if a deal is not reached in time for it to go ahead on Monday as planned.
The PM told the backbench 1922 Committee on Monday that a deal with the DUP would not affect power-sharing talks in Northern Ireland or LGBT rights.
Mrs Foster has rejected suggestions that the prospective agreement could undermine a return to power-sharing arrangements at Stormont amid claims from political rivals that the Government's stated impartiality as a mediator would be fatally undermined.
Meanwhile, Sinn Fein's Gerry Adams turned Mrs May's own slogan against her to brand it "a coalition for chaos", adding: "Any deal which undercuts in any way the process here or the Good Friday Agreement is one which has to be opposed."
It is thought Mrs Foster, despite being a Brexit supporter, could seek assurances from Mrs May that she will pursue a softer exit from the EU, given Northern Ireland's 56% Remain vote and the DUP's desire not to see a return to a hard border with Ireland.
The DUP leader is almost certain to ask for greater investment in Northern Ireland as the price of a deal.