"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", a government spokesman said. The Telegraph said senior Conservatives including Foreign Secretary Boris Johnson, interior minister Amber Rudd and Brexit minister David Davis were taking soundings over whether to replace her.
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".
The Conservatives lost their parliamentary majority in Thursday's election - a vote May called in a bid to strengthen her mandate ahead of exit talks with the EU.
Conservative legislator Anna Soubry said she should "consider her position", while another, Heidi Allen, said she may not last six months.
Without a majority, she could be forced to seek consensus on the approach she takes, potentially by performing a U-turn on single market membership, protecting the economy at the expense of immigration controls.
But Graham Brady, who chairs the influential 1922 Committee of backbench Conservative lawmakers, said a "self-indulgent" party leadership campaign would only cause more uncertainty.
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This is seen as a vote of confidence, and if it is not passed then the government will fall and Mr Corbyn will get his opportunity to form a government and purse his socialist policies in power.
DUP leader Arlene Foster said there had been "very good discussions" so far on how her 10 MPs could support a Conservative minority government, and she would travel to London to meet May on Tuesday. That means the DUP would back the government on confidence motions and budget votes, but it's not a coalition government or a broader pact.
There were also misgivings about relying on the DUP, which strongly opposes same-sex marriage and abortion.
This would give the Tories the numbers to pass a Queen's Speech on Monday 19 June, which will set out their legislative agenda.
"There's a possibility of voting the Queen's Speech down and we're going to push that all the way".
"I think everybody wants to see an agreement in the end that does respect what the British people voted for previous year - makes sure that our cooperation with Europe continues, our trade with Europe continues, our security cooperation with Europe continues", he said.
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