Democratic Unionist leader Arlene Foster is set to hold talks with Theresa May in Downing Street to finalise an agreement on propping up her minority government.
The DUP confirmed that Mrs Foster would be going to No.10 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."
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.
The 10 DUP MPs 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 deal would mean them backing the Government on its Budget and confidence motions, but could potentially lead to other issues being decided on a vote-by-vote basis.
The talks were in line with Mrs Foster's "commitment to explore how we might bring stability to the nation at this time of great challenge", her party said in a statement.
It continued: "The talks so far have been positive. Discussions will continue next week to work on the details and to reach agreement on arrangements for the new Parliament."
Number 10 had earlier said: "We can confirm that the Democratic Unionist Party have agreed to the principles of an outline agreement to support the Conservative Government on a confidence and supply basis when Parliament returns next week."
Following talks between Mrs May and the DUP on Saturday night, a second statement confirmed that no final deal had been reached.
A Downing Street spokeswoman said: "The Prime Minister has tonight spoken with the DUP to discuss finalising a confidence and supply deal when Parliament returns next week.
"We will welcome any such deal being agreed, as it will provide the stability and certainty the whole country requires as we embark on Brexit and beyond.
"As and when details are finalised both parties will put them forward."
Mrs May needs support in Parliament because the Queen of England'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.
Appearing on Sky's Sophy Ridge on Sunday, Conservative Karen Bradley said "conversations are ongoing" with the DUP when pressed on whether a deal has been struck.
"The DUP and us have voted together on many occasions in the past," the Culture Secretary told the show.
"We are two separate parties with different manifestos, but we do have common cause in some areas."
Quizzed on why the statement was sent out by Downing Street, suggesting an agreement, Ms Bradley would not be pinned down and said she did not want to "get into process points".
She said it is important to have the discussions and negotiations so that a confidence and supply arrangement can be agreed, so that the Tories can "govern in the national interest".
Also appearing on the Sky show, former minister, Anna Soubry, when asked about how she felt towards the arrangement, said: "I have friends in the DUP that I get on well with.
"I don't agree with their policies on abortion or gay rights at all, in fact I absolutely detest those policies.
"But, I think we can come together for the national good - looking especially at the economy.
"There is nothing wrong with doing that."
She said the "national interest must be the primary interest", and that stability is needed as Brexit negotiations begin.
"If the DUP are prepared to vote for a Queen's Speech, that is a sensible Queen's Speech and a good budget, I don't have difficulty with that," she said.
"I don't have to sign up to all their other stuff, which I don't agree with, at all."