Senior members of the ruling Conservative party had reportedly made the departure of Nick Timothy and Fiona Hill a condition of allowing May to stay on after losing her parliamentary majority in Thursday's vote.
He acknowledged that the government would be unable to get numerous measures promised in its election platform through Parliament.
Primer Minister Theresa May leaves Downing Street on her way to Buckingham Palace in London after Britain's election.
Former chancellor George Osborne said Mrs May's days in Downing Street were clearly numbered. Mrs May called the snap election to win a clear mandate for her plan to take Britain out of the EU's single market and customs union in order to cut immigration.
"I don't think throwing us into a leadership battle at this moment in time, when we are about to launch into these hard negotiations, would be in the best interests of the country", Evans said.
"I can still be prime minister", Corbyn said.
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. That means the DUP will back the government on key votes, but it's not a coalition government or a broader pact.
Regardless of Fallon's claims, a growing number of Conservatives are said to be "absolutely furious" with May for putting their jobs on the line for the sake of strengthening her hand overseas in Brexit talks.
Thomas Pesquet, heureux de son retour sur Terre
Thomas Pesquet est le français ayant passé le plus de temps dans l'espace avec 196 jours dans la station. L'atterrissage devrait s'effectuer au milieu de la steppe kazakhe aux alentours de 16 h (Paris).
We have had a political crisis in Northern Ireland since the collapse of the executive previous year over a heating scandal implicating Arlene Foster, the leader of the DUP.
I know that Mrs May is desperate to find some way to cling on to power in Westminster, but I appeal to her to reconsider doing so propped up by one side from Northern Ireland politics.
The DUP and Irish nationalists Sinn Fein are due to restart talks to form a new power-sharing government in Northern Ireland and avoid devolved power reverting to the British parliament in London for the first time in a decade. The DUP is a socially conservative group that opposes abortion and same-sex marriage and had links to Protestant paramilitary groups during Ireland's sectarian "Troubles". "If Mrs. May depends on the DUP...to form a government, it will be impossible for it to be even-handed".
The Labour leader, who survived an attempt to oust him by his own MPs before the election, now expects that he will be able to attract some of the party's biggest names to serve on his front bench. 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.
He said there was a "possibility" of voting down the Queen's Speech and Labour were going to "push all the way" to achieve that.
The debate on the Queen's Speech usually lasts about five or six days.
May earlier on Saturday lost her two closest aides as she struggled to reassert her leadership after a crushing election setback.
The Times of London said in an editorial that "the election appears to have been, among other things, a rejection of the vague but harshly worded prospectus for Brexit for which Mrs".
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