The DUP has reportedly said that anything is possible in Westminster talks, except a coalition with Labour leader Jeremy Corbyn. But there are significant questions about the DUP's hardline stance on gay rights and other issues, including abortion and climate change.
The British pound dropped sharply on Thursday evening after the exit poll suggested that no party will win a majority in the House of Commons after 2017's general election.
The DUP holds just 10 seats in the British parliament but this would be enough to give May's Conservatives a working majority.
The night was marked by a collapse in Ukip support and a rash of high-profile losses for the SNP, as British politics returned to a two-party system on the greatest scale since the 1970s.
A Number 10 spokesman said: 'We can confirm that the Democratic Unionist Party have agreed to the principles of an outline agreement to support the Conservative Government on a confidence and supply basis when Parliament returns next week.
Concerns over the potential impact on the upcoming Brexit negotiations and May's future as Prime Minister are now being raised, with one minister telling BBC's Laura Kuenssberg it was "hard to see how she could stay after these results".More news: Mobile will offer Apple's latest iPad Pro starting next week
It also asked for "constitutional stability for Northern Ireland", an apparent reference to efforts by Irish nationalists to seek a referendum on splitting from the United Kingdom and joining the Irish Republic. The DUP's MPs have already formed alliances with Tory backbenchers, who have been campaigning against moves to prosecute former soldiers involved in incidents in the Troubles, stretching back to the early 1970s.
The party from Northern Ireland, led by Arlene Foster, is the most likely party to team up with the Conservatives, who have won the most seats.
"She seems to have found it very hard to kind of do the baby-kissing part of campaigning".
Brexit Minister David Jones said he supported Mrs May but it was "impossible to say" if she would still be Prime Minister in six months' time. She triggered Article 50 and then called for the snap election, probably in mind that she could make it all about Brexit but that plan backfired drastically as people were concerned over the public services such as the NHS and police. "May fights to remain PM", said the front page of the Daily Telegraph, while the Times of London said: "May stares into the abyss".
The party was founded in 1971 by firebrand preacher Ian Paisley, and just as Sinn Fein has very strong links to republican paramilitary groups, the DUP has strong, historical links to loyalist paramilitary groups.
In another sign of the dangers facing Mrs May, Sunday papers reported that Boris Johnson was either being encouraged to make a leadership bid in an effort to oust her, or actually preparing one - a claim dismissed as "tripe" by the Foreign Secretary.