The moves buy May a temporary reprieve.
She said Brexit talks would begin on June 19 as scheduled, the same day as the formal reopening of parliament.
One prominent Conservative said that if the party emerges with any fewer seats than it held before the election, Mrs May would be "in trouble".
The DUP could agree to back the Government on the budget and any vote of confidence while deciding other measures on a vote-by-vote basis.
Earlier, Downing Street had said a preliminary agreement had already been secured. As May struggled to contain the fallout, her two closest aides resigned.
Mrs May is seeking to rely on the Democratic Unionist Party's 10 MPs to win key votes in Parliament.
"We cant go on with a period of great instability", he said, adding that the Labour Party was ready to fight another election campaign.
With one seat left to declare, the Tories are eight seats short of the 326 figure needed to command a majority and must relay on the DUP to continue to rule.
Meanwhile, Labour - which had been written off by critics as all but unelectable - surged to 261 seats, up 29 from its tally in the 2015 election.
Before the election, she had been widely expected to sack finance minister Philip Hammond following a reported clash over her Brexit strategy.
May's campaign catchphrase of a "strong and stable government" has backfired.
Campaigning was suspended twice - another unprecedented development for a United Kingdom election, according to Simon Usherwood of the University of Surrey, who said that the attacks could revive the trauma of IRA bombings of the 1980s.
"It was a disaster", he said.
Co-chief of staff Mr Timothy was widely blamed for the disastrous proposal to reform social care funding in the party's manifesto without consultation.
The bloodshed thrust the issue of security into the heart of the debate, with Corbyn even calling on May to step down over a major reduction in police forces during the six years she was interior minister. If accurate it suggests May's gamble to call a snap election prior to Brexit negotiations was ultimately a horrendous mistake.
"What the country needs more than ever is certainty", she said.
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".
In a statement outside Downing Street, the 60-year-old premier promised to "fulfil the promise of Brexit".
The single biggest issue confronting Theresa May in her in-tray.
The alliance makes some modernizing Conservatives uneasy.
Scottish Conservative leader Ruth Davidson has since sought assurances from May that any deal with the DUP would not affect LGBTI (Lesbian, Gay, Bisexual, Transgender, and/or Intersex) rights across the UK.
"Absolutely", 68-year-old Corbyn told the Sunday Mirror, still not giving up his dream of becoming the next British Prime Minister.
"I don't think we should talk about some prolongation of the deadline", he said in Prague. "When it becomes a matter for me is when people try to redefine marriage".
Still the deal with the DUP risks upsetting the political balance in Northern Ireland.
Jeremy Corbyn, the Labour leader, has urged May to resign as he said she should "go and make way for a government that is truly representative of this country". But we will act in the national interest and do what it right for the United Kingdom as a whole and in particular for Northern Ireland.
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