South Staffordshire MP Gavin Williamson working on deal with DUP for Theresa May

It comes as the Prime Minister's two closest aides quit in the wake of the disastrous General Election result.

Mrs May's joint chiefs of staff, Birmingham born Nick Timothy and Fiona Hill resigned in the wake of the election which saw Mrs May lose her Commons majority.

The Prime Minister looks set to be backed by the DUP as she attempts to run a minority government after Chief Whip Mr Williamson was sent to Belfast to conduct the negotiations.

The DUP confirmed that Mrs Foster would be going to No10 on Tuesday after discussions in Belfast over the weekend were said to have made “good progress”.

Mrs Foster told Sky News: “We had very good discussions yesterday with the Conservative Party in relation to how we could support them in forming a national government – one that would bring stability to the nation. Those discussions continue.”

Downing Street initially said on Saturday that an outline agreement on a “confidence and supply” arrangement had been reached with the DUP which would be put to the Cabinet for discussion on Monday.

But it later disclosed that no deal had been finalised and talks on the arrangement will continue during the week as Mrs May desperately tries to shore up her position after losing her Commons majority in the election.

The strength of any deal looks set to be tested when the Commons meets, 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.”

A "confidence and supply" arrangement is a far looser deal than a formal coalition or pact.

It would mean the DUP backing the Government on its Budget and prevent it being brought down by motions of no confidence, but could potentially lead to other issues being decided on a vote-by-vote basis.

Nationally, the Tories won 318 seats to Labour's 262 - eight short of a majority.

Most of the major parties had ruled out a formal coalition with the Tories until the DUP stepped forward.

It is understood that Mr Williamson, appointed Chief Whip last July, has been asked to negotiate a formal coalition with DUP, similar to the one brokered by the Tories and Liberal Democrats in 2010.

Mr Williamson has been opposed to the confidence and supply arrangement, which essentially sees the smaller party back the larger party on key issues - in exchange for some key demands of their own.

The 40-year-old says he remains '100 per cent committed' to Theresa May.

The Prime Minister has made clear she wants support from her "friends and allies" in Northern Ireland's DUP to secure her minority administration ahead of the Queen's Speech on June 19.

Meanwhile, 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.

Mr Timothy acknowledged that one of his regrets was the way Mrs May's social care policy - dubbed the "dementia tax" - by critics had been handled.

The Prime Minister was forced to perform an unprecedented U-turn within days of the publication of the Tory manifesto by announcing that there would be a cap on social care costs, something that had been absent in the original policy document.

In a resignation message on the ConservativeHome website, 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 I would like to make clear that the bizarre media reports about my own role in the policy's inclusion are wrong: it had been the subject of many months of work within Whitehall, and it was not my personal pet project.

"I chose not to rebut these reports as they were published, as to have done so would have been a distraction for the campaign. But I take responsibility for the content of the whole manifesto, which I continue to believe is an honest and strong programme for government."

Mrs May was expected to make further appointments to her Cabinet on Saturday, but the damage to Mrs May's standing makes it less likely she will risk alienating colleagues by carrying out an extensive reshuffle as she cannot afford to have disgruntled former ministers sniping at her from the backbenches.

After speculation the PM would use a solid win in the election to move Philip Hammond from the Treasury, he and other potential successors as Tory leader, Foreign Secretary Boris Johnson and Home Secretary Amber Rudd, remained in place.

With Brexit Secretary David Davis and Defence Secretary Sir Michael Fallon also staying put, there were suggestions changes could just centre on replacing the eight ministers who lost their seats as the Tory Commons tally fell to 318.

There was also unease within the party about the link-up with the DUP, which strongly opposes same-sex marriage and abortion.

Scottish Tory leader Ruth Davidson said she had demanded a "categoric assurance" that gay rights would not be affected by a deal with the DUP, which strongly opposes marriage equality.

Ms Davidson, who became engaged to partner Jen Wilson in May 2016, told the BBC: "I was fairly straightforward with her (Mrs May) and I told her that there were a number of things that count to me more than the party.

"One of them is country, one of the others is LGBTI rights.

"I asked for a categoric assurance that if any deal or scoping deal was done with the DUP there would be absolutely no rescission of LGBTI rights in the rest of the UK, in Great Britain, and that we would use any influence that we had to advance LGBTI rights in Northern Ireland."

Tory MP Tom Tugendhat said: "I joined a party that introduced equal marriage, backs civil rights and defends freedom of faith. Those principles won't be compromised."

In an indication of the opposition to Mr Timothy and Ms Hill, former minister Anna Soubry called for them to be sacked while Mrs May's former communications chief Katie Perrior, who left Downing Street when the election was called, hit out at their "rude, abusive, childish behaviour".

Writing in The Times she said: "Mrs May condoned their behaviour and turned a blind eye or didn't understand how destructive they both were."

Amid reports that senior Tories were sounding out potential replacements for Mrs May, prominent Conservative MP Heidi Allen said the Prime Minister had six months at most left in Downing Street.

But former leader Lord Hague, writing in the Daily Telegraph, warned against a leadership contest: "Voters do not want further months of uncertainty and upheaval."

Labour deputy leader Tom Watson said Mrs May was responsible for the election result, which saw her party fall short of a Commons majority.

He tweeted: "The adviser takes the fall but Theresa May is the one responsible for her own defeat."