Former Conservative party leaders have warned any immediate leadership challenge would be too disruptive, but most commentators believe May can not survive in the long term.
The moves buy May a temporary reprieve. This, in effect, will be a vote of confidence in Theresa May's government and the first test of any deal with the DUP.
Downing St. said the Cabinet will discuss the agreement on Monday.
May has shown little public contrition for the electoral gamble that backfired but was forced to accept the resignations of her two closest aides - reportedly a requirement by cabinet colleagues for allowing her to stay in office.
"I look forward to meeting with him again tomorrow if his appointment is reaffirmed, but yes I think it's an important issue that you raise - the objectivity of both governments, and both governments working strictly in accordance with our legal responsibilities under the Belfast Agreement, the Irish government as co-guarantor, indeed the British Government as co-guarantor".
Some senior Tories had made the removal of Hill and Timothy a condition for continuing to support May, who has vowed to remain prime minister.
Labour gained 30 seats in the general election to take its total to 262 seats, but the Conservatives remain the largest party in Parliament with 318 seats. A buoyant Corbyn says Labour is capable of forming a govenment. His party "didn't win the election", Corbyn says, but he is prepared to continue resisting the Conservative agenda - especially what's known as "hard Brexit", a clean break with the European Union and departure from its single market system. He also noted that it is the voters and mostly the young people who determine the result of the elections and not the polls.
Mr Corbyn also claimed Ms May's plans to complete Brexit through a "Great Repeal Bill" were "history" and would have to be dropped - although he insisted the United Kingdom would leave the EU.
Foster has yet to set out her demands but her party wants an end to prosecutions of British soldiers who fought in Northern Ireland and an easing of restrictions on parades.
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. May's departure from Downing St.is remorseless".
"I can still be prime minister", Corbyn tells the Sunday Mirror.
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The alliance makes some modernizing Conservatives uneasy.
He stressed he did not share their ultra-conservative views on issues such as abortion and homosexuality, which have caused disquiet among many Conservatives.
The socially-conservative DUP, founded in 1971 by the late Ian Paisley (whose son Ian Paisley Jnr.is a DUP MP), won 10 seats in the general election, securing a third of all votes cast in Northern Ireland. She's hitched the Conservative Party's fortunes to the politics of the DUP, and every day that she clings to power with this arrangement she tarnishes the Conservatives' brand.
Northern Ireland is the only part of the U.K.in which same-sex marriage is illegal.
DUP Leader Arlene Foster recently denied the party was homophobic.
"But what I'm doing now is actually getting on with the immediate job".
The election result will provide the Labour Party, especially its leader Jeremy Corbyn, with a shot in the arm.
Newspaper headlines saw her as just clinging on.
May shrugged off suggestions her days in Downing St. were numbered.
"I think in this case, most pundits would agree that she really did have a bad, bad, bad campaign ..."
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