The Democratic Unionist Party has cooled talk of a done deal to prop up Theresa May in Downing Street, insisting it has yet to “enter discussions”.

Arlene Foster, DUP’s leader, said her party needed to “explore how it may be possible to bring stability to our nation at this time of great challenge”.

Nevertheless, the expectation is that a deal will be struck after Jeffrey Donaldson, the party's chief whip, said the hung parliament “puts us in a very strong negotiating position”.

In her speech in Downing Street, the Prime Minister announced she had the backing of the DUP to provide the “certainty” the country badly needed.

“Our two parties have enjoyed a strong relationship over many years, and this gives me the confidence to believe that we will be able to work together in the interests of the whole United Kingdom,” she added.

However, some saw the short speech as an attempt to bounce the Conservative party into accepting Ms May should stay in power – even as her Cabinet remained silent.

Some Conservative MPs believe her position is untenable. Overnight, George Osborne, no longer an MP, said: “Personally I don’t see how she can survive for the long term.”

And, earlier, Ms Foster herself doubted Ms May could stay in No 10, telling the BBC: “I don't know”, adding: “I think it will be difficult for her to survive.”

Later, at a news conference, Ms Foster said the DUP would go into talks with the Conservatives, with the future of the Union as “our guiding star”.

“We may represent Northern Ireland constituencies in the House of Commons, but we are as seized of the interests of the United Kingdom as a whole, as we are for Northern Ireland.

“The Prime Minister has spoken with me this morning and we will enter discussions with the Conservatives to explore how it may be possible to bring stability to our nation at this time of great challenge.”

With the DUP’s support, Ms May would enjoy an effective majority of at least 12, but faces big hurdles to impose her programme and to reassert her battered authority.

Underlining the extraordinary make-up of the new government, she would be the first Prime Minister to rely on Irish MPs for their majority since the Liberal Herbert Asquith, way back in 1910.

There will be questions asked about the DUP’s opposition to same-sex marriage and abortion and its flirtation with denying climate change.

Although the DUP campaigned for Leave, Ms Foster has spoken out against a “hard Brexit”, to maintain a “frictionless border” with the Irish Republic in the vital interests of trade.

Tim Farron, the Liberal Democrat leader said an agreement with the DUP would have “profound implications” for both Brexit and the peace process in Northern Ireland.

“It is clear that the Prime Minister has done a deal with the DUP. She must now make clear what the terms of that deal are. The British people have a right to know,” he said.

Some in Northern Ireland worry about the impact on the all-party talks to restore devolution in Belfast, which are scheduled to restart on Monday.