May is seeking a deal with a Northern Irish party to prop up the Conservative minority administration, and lawmakers said the rebuff from voters meant the government will have to abandon planned policies and re-think its strategy for European Union exit talks.
DUP leader Arlene Foster said her party would be entering talks with the Conservatives to explore "how it may it be possible to bring stability to our nation at this time of great challenge".
He said: "The new Cabinet obviously will meet early next week".
There was no mention of what concessions the DUP may have asked for, amid growing concern about the influence of a party opposed to abortion and gay marriage, and which has proved hugely controversial in the past over the homophobic and sectarian views of some of its representatives.
The Opposition Labour leader, Jeremy Corbyn, has also said that his party is waiting in the wings to take charge after he defied pollsters? predictions by gaining 30 seats in Thursday's election, leaving a hung Parliament.
Speaking on the BBC's Andrew Marr show, Mr Corbyn said: "We're going to put down a substantial amendment to the Queen's Speech which will contain within it the main points of our manifesto, and so we'll invite the House to consider all the issues we put forward which I've mentioned - jobs-first Brexit, mention the issues of young people and austerity, there's many other things".
Labour's next big push will come during the Queen's Speech on June 19 which will herald the opening of a new parliament in the United Kingdom and will contain a new budget as well as the program of government.
Mr Kenny, who will formally retire as Taoiseach in the coming days to be replaced by new Fine Gael leader Leo Varadkar, said he spoke with Mrs May about protecting the Good Friday Agreement.
The following scenarios touch on what may happen now voters have dashed May's hopes for a bigger majority to negotiate and left her dependent on pro-Brexit Ulster Protestants and on political rivals reportedly eyeing their moment to oust her. Across the country, people voted Labour despite not liking Corbyn.
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As the dust settles, one issue will become the focus for investors: whether Britain is more or less likely to retain privileged access to the EU's single market.
Labour will also be pushing for more public spending on social welfare, free university education and higher taxes for the richest one percent. It will be more hard for a Conservative government to walk away from the European Union without a trade deal, as May has threatened to do.
The former minister of finance George Osborne, who stepped aside after the election informed reporters that the results were disastrous for his party.
The right-wing British press speculated Sunday that Foreign Secretary Boris Johnson, a former journalist and the former Mayor of London, was plotting a leadership coup.
"I just thought what a rubbish campaign that Theresa May ran, quite honestly, I really did".
Front and center among many questions facing May's new government is the matter of Brexit. Conservative MPs are publicly airing their anger, some calling for her ouster and others demanding radical change in her style of leadership.
UK's leading LGBTI charity Stonewall, condemned the coalition between the Conservatives and Democratic Unionist Party (DUP) yesterday (9 June). "I think we'll know very shortly".
The Prime Minister explained that she is working towards a confidence and supply deal with the DUP which would provide stability and certainty for the United Kingdom going forward. "We are willing to serve the country", said John McDonnell, a senior Labour politician.
Under pressure from Conservative cabinet ministers, May accepted the resignation of her two top aides, her co-chiefs of staff Nick Timothy and Fiona Hill, on Saturday.
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