With 649 of 650 seats in the House of Commons declared, the Conservatives had 318 - short of the 326 they needed for an outright majority.
Paul Nuttall resigned as leader of UKIP, the party having failed to win any seats and losing a substantial portion of its support.
Katie Perrior, May's former director of communications, who quit when the snap election was called, has attacked the prime minister's governing style, and her joint chiefs of staff, Fiona Hill and Nick Timothy.
She didn't get it, in fact she lost the majority that Cameron's leadership had given the Conservative Party, but stays on in power propped up by the Democratic Union Party, claiming that the party she leads in the only one that has legitimacy to govern.
"It is quite possible there will be an election later this year or early next year and that might be a good thing because we can not go on with a period of great instability", he told the BBC.
The DUP whose 10 seats would allow the government to get measures through Parliament, is a socially conservative pro-British Protestant group that opposes abortion and same-sex marriage and includes both environmentalists and climate-change deniers among its senior ranks.
In response to the earlier Downing Street statement, Liberal Democrat leader Tim Farron called on Mrs May to make public immediately the details of the deal "stitched up behind closed doors".
DUP agrees deal to support Conservative government
They added: "Her [Mrs May's] lack of authority, her lack of credibility, how she doesn't have a mandate, that's the priority". Davis also said walking away without securing a deal with the remaining 27 European Union states remained a possibility.
Thursday's United Kingdom general election saw the Conservatives lose more seats than expected in Parliament, while the Labour gained more seats than expected.
Foster said Friday it would be "difficult" for May to continue in her role.
Ms Davidson said she had received an assurance from the PM there would be no erosion of gay rights in Great Britain. However, she has made the mistake of forming the government and then going to the DUP, which in negotiating terms says to the other side of the table "we need you more than you need us".
Brexit talks are set to begin in a week and the Tories will now have to take into consideration the views of the coalition partner and arrive at a consensus before it enters into deliberations with the European Union on the exit.
The new parliament will be sworn in Tuesday, but the real test for May is likely to come on June 19, when MPs are to vote on her programme after it is outlined in parliament by Queen Elizabeth II on June 19.
Before leaving for the Palace on Friday morning, to have an audience with the Queen before attempting to form a government, the Conservative leader is said to have been crying.
A buoyant Jeremy Corbyn, however, said in an interview with the Sunday Mirror: "I can still be Prime Minister". If some Conservative MPs desert the party on key votes than May's plans would be left in tatters.
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