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.
British Prime Minister Theresa May was seeking a deal with a small Northern Irish party on Sunday to stay in power after losing her party's parliamentary majority in a catastrophic electoral gamble just days before Brexit talks are set to start.
Jeremy Corbyn, the Labor Party leader often dismissed (even by many in his own party) as an ineffectual ideologue, galvanized younger and working-class voters with an egalitarian message of "For the Many, Not the Few".
Their departures were seen as a Downing Street bid to stave off a far more dramatic resignation: that of the prime minister herself.
However, former Conservative party leaders warned against any immediate change, with Iain Duncan Smith saying leadership contest would be a "catastrophe".
Johnson, one of the Conservatives' most popular politicians, tweeted that an article in the Mail on Sunday newspaper headlined "Boris set to launch bid to be PM as May clings on" was "tripe".
As Britain took stock of the stunning results of a snap election that wiped out the parliamentary majority of Prime Minister Theresa May's Conservative Party, one narrative bubbled up to the surface: The youth had spoken. Labour surpassed expectations by winning 262.
The Labour leader said the current Tory top team lack the "credibility" to lead following Thursday's general election and compared them to the 1970s Ted Heath government.
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.
"I am a man of my word so what I'm going to do is just sit here and eat my book while you guys carry on".
Graham Brady, who chairs the influential 1922 Committee of backbench Conservative lawmakers, said a "self-indulgent" party leadership campaign would only cause more uncertainty.
The new parliament will be sworn in Tuesday, but the real test for May is likely to come on June 19, when MPs are to vote on her programme after it is outlined in parliament by Queen Elizabeth II on June 19.
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May called the election to win explicit backing for her stance on Brexit, which involves leaving the EU's single market and imposing restrictions on immigration while trying to negotiate free trade deal with the bloc. This is expected to have an impact on the Brexit negotiations, which are due to open with the European Union on June 19.
Some say her failure means the government must now take a more flexible approach to the divorce, potentially softening the exit terms.
On Brexit, the DUP supports leaving the European Union but opposes a return to a "hard" border with Ireland - which could happen if May carries out her threat to walk away from talks rather than accept a "bad deal".
"We have made good progress but the discussions continue", said DUP leader Arlene Foster.
"The Conservatives have not yet broken the British system of democracy, but through their hubris and incompetence they have managed to make a mockery of it", it said in an editorial.
Conservative lawmaker Nicky Morgan told ITV that she could support a confidence and supply arrangement with the DUP, but any closer deal would be "a step too far".
Such a move would be strongly opposed by the Irish nationalists of Sinn Fein who share power in Belfast under a 1998 peace agreement that ended three decades of violence between nationalists and pro-British loyalists.
The British government doesn't have long to ink a deal. It is due to present its platform for the next session in the Queen's Speech at the State Opening of Parliament on June 19. The speech will be followed by several days of debate and a vote. There's a possibility of voting it down and we're going to push that all the way.
He added: "We will - obviously - amend the Queen's Speech". "Everybody wants to see an agreement in the end that does respect what the British people voted for past year", he said.
Jeremy Corbyn still thinks he can become prime minister instead of Theresa May.
"It's just how long she's going to remain on death row", former Conservative finance minister George Osborne, who was sacked by May when she became prime minister previous year, told the BBC.
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