Theresa May looks set to be backed by the Democratic Unionist Party as she attempts to run a minority government following her General Election disaster.
Downing Street said an outline agreement on a "confidence and supply" arrangement had been reached which will be put to the Cabinet for discussion on Monday.
But the strength of that deal looks set to be tested when the Commons meets next week, with Jeremy Corbyn vowing to try to bring down the Government by defeating Mrs May in Parliament and insisting: "I can still be prime minister."
The 10 MPs from the Northern Irish party could prove crucial in supporting the Conservatives on key votes after Thursday's election saw Mrs May lose control of the Commons.
A "confidence and supply" arrangement is a far looser deal than a formal coalition.
It would mean the DUP backing the Government on its Budget and confidence motions, but could potentially lead to other issues being decided on a vote-by-vote basis.
Mrs May needs to shore up her position in Parliament because the Queen's Speech setting out the Government's programme is due on June 19, with a highly significant vote on its content expected after a few days' debate.
Mr Corbyn, whose Labour party won 262 seats in the election, told the Sunday Mirror he would oppose the Queen's Speech all the way.
"I can still be prime minister. This is still on. Absolutely," he said.
"Theresa May has been to the palace. She's now attempting to form a government. She's then got to present a programme to Parliament.
"There's a possibility of voting the Queen's Speech down and we're going to push that all the way."
The potential deal with the DUP came after Mrs May sent her Chief Whip Gavin Williamson to Belfast for talks after the election left the Tories eight seats short of the 326 required for a majority.
On a dramatic day:
:: Mrs May's two closest aides Nick Timothy and Fiona Hill quit after coming under intense pressure from Tories following the election result.
:: Ex-minister Gavin Barwell, who lost his seat in Thursday's election, was named as the new chief of staff, replacing Mr Timothy and Ms Hill.
:: Scottish Tory leader Ruth Davidson indicated she wanted to see a fresh approach to Brexit by seeking a consensus across parties.
:: A former minister said Mrs May must not be allowed to lead the party into the next election.
The role of Mr Timothy and Ms Hill as Mrs May's joint chiefs of staff had been severely criticised by disgruntled Tories in the wake of the election result.
There were also misgivings about relying on the DUP, which strongly opposes same-sex marriage and abortion.
In a resignation statement on the ConservativeHome website, Mr Timothy acknowledged one of his regrets was the way Mrs May's social care policy, dubbed the "dementia tax" by critics, had been handled.
Mr Timothy said: "I take responsibility for my part in this election campaign, which was the oversight of our policy programme.
"In particular, I regret the decision not to include in the manifesto a ceiling as well as a floor in our proposal to help meet the increasing cost of social care."
But he also set out his concerns about the way the campaign was managed, in comments viewed as a sideswipe at election guru Sir Lynton Crosby.
The campaign failed to get "Theresa's positive plan for the future across" or "notice the surge in Labour support".
In her statement Ms Hill said: "I have no doubt at all that Theresa May will continue to serve and work hard as Prime Minister and do it brilliantly."
Mrs May called the election claiming she wanted a stronger hand in the forthcoming Brexit negotiations, due to start on June 19.
Without a majority, she could be forced to seek consensus on the approach she takes, as Scottish Tory leader Ms Davidson pointed out.
Ms Davidson, who wields considerable influence after the Scottish Conservatives won 13 seats, said: "I want to ensure that we can look again at issues like Brexit which we know we are now going to have to get cross-party support for.
"And move to a consensus within the country about what it means and what we seek to achieve as we leave."
In a hint that curbing immigration could become a lower priority than safeguarding the economy she said: "It is about making sure that we put free trade at the heart of what it is we seek to achieve as we leave."
Mrs May was working on a Cabinet reshuffle, although the election result makes it less likely she will risk alienating colleagues by making wholesale changes as she cannot afford to have disgruntled former ministers sniping at her from the backbenches.
Former leader Lord Hague, writing in the Daily Telegraph, warned against a leadership contest, saying: "Voters do not want further months of uncertainty and upheaval."
Ex-minister Lord Barker said there was little appetite for a contest but told BBC's Newsnight: "There's clearly not going to be another election with Theresa May at the head.
"We have discovered that she is a competent minister, potentially a tough negotiator, but a terrible campaigner."
In a sign political developments in the UK were being closely watched in Brussels ahead of the Brexit talks, Martin Selmayr, senior aide to European Commission president Jean Claude-Juncker, likened the aides' resignations to the sacrifice of a pawn in the game of chess, using the German word "bauernopfer" in a tweet.
Liberal Democrat leader Tim Farron said: "Theresa May needs to make the terms of her confidence and supply deal with the DUP clear to the British people immediately.
"The actions of this Government will have profound implications for the Brexit negotiations and the future of our country.
"At such a critical time, the Prime Minister must be clear with the people about the deal she has stitched up with the DUP behind closed doors."