Theresa May is expected to finalise her team of ministers later as she seeks to form a government with the support of the Democratic Unionist Party.
Ministers say initial talks have begun with Northern Ireland’s DUP after the Conservatives failed to secure a majority in Thursday’s election.
The Tories needed 326 seats to win but fell short by eight. The DUP won 10.
Labour has urged Mrs May to “make way” for it to form a government and she also faces criticism over the DUP deal.
The leader of the Scottish Conservatives, Ruth Davidson, tweeted a link to a speech she had made about same-sex marriage – something the DUP opposes.
A number of high-profile government figures have already confirmed they are keeping their jobs in the wake of an election that saw the Tories lose 13 seats.
Philip Hammond will stay as Chancellor, Boris Johnson will remain as Foreign Secretary and Amber Rudd – narrowly re-elected as an MP after a recount – will continue as Home Secretary.
David Davis will also stay on as Brexit Secretary and Sir Michael Fallon will keep his role as Defence Secretary.
But eight ministers lost their seats at the general election and will need to be replaced.
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Mrs May announced her intention to work with the DUP to form a government on Friday, just hours after an election that has left the UK with a hung parliament for the second time in seven years.
Speaking on the steps of No 10 after an audience with the Queen, the prime minister said the parties had a “strong relationship” and that she intended to form a government which could “provide certainty and lead Britain forward at this critical time for our country”.
DUP leader Arlene Foster confirmed that she had spoken to Mrs May and that they would speak further to “explore how it may be possible to bring stability to this nation at this time of great challenge”.
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DUP will use leverage on May deal
Analysis by political correspondent Gary O’Donoghue
The clock is ticking for Theresa May. She needs to conclude a deal with the DUP in the next week or so ahead of the Queen’s Speech, which will set out the new government’s agenda.
That takes place on Monday 19 June – the same day Brexit negotiations are due to start.
The DUP and its 10 MPs are in a very strong position. It’s all their Christmases rolled into one and they will make sure they leverage as much as they can from their advantage.
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Money for Northern Ireland will undoubtedly be part of their demands, and Mrs May will expect that. But trickier will be any demands they have about the implementation of Brexit in Northern Ireland – in particular the DUP’s determination to maintain a soft border with the south.
Another potential problem is the planned restart of negotiations for power-sharing in the province.
Typically the British government tries to act as an honest broker between Republicans and Unionists. But if Mrs May is doing a deal with the DUP, that could make it harder to reach an agreement with Sinn Fein.
Mrs May’s decision to seek a deal with the DUP has prompted concerns from some Tories.
Northern Ireland is the only part of the UK where same-sex marriage is not legal.
Ruth Davidson, who is gay, plans to marry her partner in the near future and said she had been “straightforward” with Mrs May about her concerns.
“I told her that there were a number of things that count to me more than the party,” she told the BBC. “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.”
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By winning 12 additional seats in Scotland, Ruth Davidson played a significant part in helping Theresa May to stay in Downing Street, BBC Scotland editor Sarah Smith says.
Ms Davidson clearly plans to use her new-found influence to try to affect the Brexit negotiations as well, suggesting that she believes the UK should try to remain in the EU single market, our editor adds.
Other members of the party have criticised Mrs May for staying on in Downing Street after failing to secure a majority government.
Former Business Minister Anna Soubry called for her to “consider her position” after a “disastrous” election campaign.
Labour has also demanded her resignation, with its leader Jeremy Corbyn saying Mrs May should “make way” for a government that would be “truly representative of the people of this country”.
The party won 262 seats in the election – up by 30 from 2015.
With 40% of the vote, it also secured its biggest vote share since the 2001 election when Tony Blair won his second term as PM.
Mr Corbyn said his party was ready to form a minority government of its own, but stressed he would not enter into any “pacts or deals” with other parties.
Mr Corbyn, who is expected to announce his shadow cabinet on Sunday, added: “We are ready to serve the people who have put their trust in us.”
Speaking on BBC’s Question Time, Mr Grayling defended the prime minister, saying she needed to stay in the role for the “foreseeable future”.
“Not only must she not resign, she has to not resign in the interest of the country because we need to move forward, we have got to go into the Brexit negotiations,” he added.
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