With all but one of 650 seats declared, the centre-right party had the most votes and the most seats but lost its majority in the House of Commons.
"She is prime minister, she remains prime minister and the country has to come first", said Ian Duncan Smith, a former Conservative Party leader and one of the most prominent campaigners to leave the European Union.
She would stay on as prime minister.
Business, already struggling with the uncertainties of the two-year Brexit negotiating process, urged party leaders to work together.
But rumors swirled of plots to oust May. There were media reports that Johnson and other leading party members were weighing party leadership challenges.
She has led a Conservative resurgence in Scotland, providing crucial backing to May's slender majority and dealing a severe blow to the separatist Scottish National Party (SNP), whose number of seats plunged to 35 from 56. Labour surpassed expectations by winning 262. With Thursday's election providing 29 new Labour MPs, Corbyn's energetic election campaign has paid off. He later said it was it was "pretty clear who has won this election".
He acknowledged that the government would now be unable to get numerous measures promised in its election platform through Parliament.
May unexpectedly called the snap election seven weeks ago, three years early, polls predicting she would massively increase the slim majority she inherited from Cameron.
Full Liberation of Mosul Imminent as Noose Tightened on ISIS
ISIL has lost much of that territory over the last three years, and Mosul is their last urban bastion in Iraq. The US-backed military offensive to recapture the Iraqi city entered its ninth month on Friday.
May confirmed she meant to start talks with the Europeans on June 19 as planned, promising to "get to work".
Front and center among many questions facing May's new government is the matter of Brexit. He said voters had rejected her Brexit plan.
At a press conference today, DUP leader Arlene Foster was vague about a working relationship with May's Conservatives.
"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. 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".
She did not, however, provide details of the nature of the alliance with the DUP.
Because the DUP are a party that only stand for election in Northern Ireland, you may not have heard of them unless you live there.
Irish Prime Minister Enda Kenny tweeted Sunday that he had spoken with May "and indicated my concern that nothing should happen to put (the Good Friday Agreement) at risk".
They'll be hoping to influence the government's plans for Northern Ireland, and to get a better deal for people there. Since any deal is not expected to involve a formal coalition, such talks may not hold up the formation of government. The speech will be followed by several days of debate and a vote.
Labour has said it will not seek a coalition, instead seeking to govern as a minority government if possible. "She's attempting to form a government", Corbyn told the Mirror.