Earlier, May's former head of communications launched a series of brutal attacks on the prime minister's decision to place so much trust in her two chiefs of staff and her inability to connect with voters or the rest of the party.
The DUP won 10 seats, which would mean the Conservatives would have support from a majority of 328 deputies if they confirm the confidence-and-supply arrangement with the DUP.
The timing is challenging, with Britain due to start negotiating the terms of its exit from the European Union with the bloc's 27 other members on June 19.
Aware that the opposition Labour party had made election gains by focusing heavily on national issues, while the Conservatives drew on Brexit, May listed areas such as education and housing as top priorities.
Former Treasury chief George Osborne - who was sacked by May previous year - called May a "dead woman walking", and opposition leader Jeremy Corbyn said he was ready to contest another election at any time.
Mr Corbyn said the election signified a rejection of the "politics of fear" and said "we are ready any time" to govern, should Ms May's premiership collapse. But few believe she can hang on for more than a few months.
"I don't think she does have a majority in the House of Commons for leaving the single market", said Anna Soubry, a Conservative member of parliament who campaigned ahead of last year's referendum for Britain to stay in the EU.
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.
But there were early signs that without a parliamentary majority, a weakened May could not count on all of her party's lawmakers to support her approach. "I think we'll know very shortly". One is for us to get on, in what I think is the responsible way, and try to form a government and try to offer the right kind of responsible leadership that the country needs.
To stay in power, the Conservatives are seeking support from Northern Ireland's Democratic Unionist Party.
May, whose two key aides resigned amid anger about how they handled the campaign, leads a brittle government as the United Kingdom prepares to start talks on exiting the European Union. DUP leader Arlene Foster told Sky News she would be meeting May in Downing Street tomorrow.
Downing Street said it hopes to finalize the deal next week, after Parliament resumes sitting.
Defence Secretary Michael Fallon said the government was not looking at a formal coalition but would seek assurances that the DUP would vote with May "on the big things" such as the budget, defence issues and Brexit.
The arrangement with the DUP will make governing easier, but it makes some Conservatives uneasy. Still the deal with the DUP risks upsetting the political balance in Northern Ireland.
"I can still be prime minister".
The 1998 Good Friday agreement set up power sharing in Northern Ireland, largely ending years of sectarian violence. It aligns London more closely with the pro-British side in the divided province, where a power-sharing government with Irish nationalists is suspended. Without the amendments, he said Labour would try to vote down the speech.
"The prime minister has tonight spoken with the DUP to discuss finalising a "confidence-and-supply" deal when Parliament returns next week", a statement released by Downing Street late on Saturday night said. "This is still on".