Downing Street said an outline agreement on a “confidence and supply” arrangement had been reached and will be put to the Cabinet for discussion on Monday.
The 10 MPs from the socially conservative Northern Irish party could prove crucial in supporting the Conservatives on key votes after Thursday's election saw Ms May lose control of the Commons.
Ms May faced calls to resign as Prime Minister amid fierce criticism for calling the snap general election after the Conservatives fell well short of their own expectations winning just 318 seats.
A total of 326 seats is needed for a majority in the Commons, and attempts to run a minority government are normally doomed to fail because of the inability to get any legislation through Parliament, meaning an arrangement with another party is necessary.
An agreement of confidence and supply means the supporting party will back the Government in motions of confidence by either voting in favour of abstaining, while retaining the right to vote otherwise in matters of conscience. "Supply" refers to bills required for a minority government to receive money to allow it to enact its policies.
It is a far looser arrangement than the coalition which the Conservatives and Liberal Democrats entered into in 2010, and the DUP will not have any of its MPs becoming government or cabinet ministers.
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.
"We welcome this commitment, which can provide the stability and certainty the whole country requires as we embark on Brexit and beyond.
“The details will be put forward for discussion and agreement at a Cabinet meeting on Monday.”
It is not known what concessions the Conservatives have agreed to in order to secure the support of the DUP, a party with religious origins and a known historical association with Unionist terrorist groups. But the DUP is thought likely to have asked for an agreement not to further cut benefits as the party is pro-welfare state and has consistently opposed the Government's austerity measures.
A coalition would have offered more stability but would have caused more controversy than the confidence and supply arrangement because of the DUP's stance on issues such as same-sex marriage.
The Conservative Chief Whip Gavin Williamson held hastily arranged talks with his DUP counterpart in Belfast today after a hung Parliament result was confirmed on the early hours of Friday morning.
The decision by the Conservatives to form such an agreement with the DUP is controversial, with more than half a million people already signing a petition opposing such an arrangement and protesters taking to the streets of London today to voice their discontent at the prospect.
Members of the DUP are known for their ultra conservative beliefs, including climate change denial, as well as for their evangelical Christian views of sexuality.
The party is led by Arlene Foster in Northern Ireland and Sir Jeffrey Donaldson in Westminster. It is the largest single party in Northern Ireland and is pro-Brexit as well as pro-Union.
With Brexit negotiations with the European Union set to begin on 19 June, Ms May immediately faced calls to make clear the terms of the agreement.
Liberal Democrat leader Tim Farron said: "Theresa May needs to make the terms of her confidence and supply deal with the DUP clear to the British people immediately.
"The actions of this Government will have profound implications for the Brexit negotiations and the future of our country.
"At such a critical time, the Prime Minister must be clear with the people about the deal she has stitched up with the DUP behind closed doors."
Ahead of the announcement, a former DUP member of the Northern Ireland Assembly said he does not expect a deal to last more than 12 or 18 months.
Alastair Ross told BBC Radio Ulster's Inside Politics programme he did not believe any agreement between the Conservatives and the DUP would be a formal arrangement.
"I would have thought the DUP would want to have as much flexibility in any arrangement as they can get, because they will not want to get tied into some of the less popular things," he said.
"They (the DUP) would perhaps support them (the Conservatives) in a budget and the Queen's Speech, but allow themselves the flexibility to take different positions to the Conservative Party if it's in the interests of Northern Ireland to do so.
"I'd be very, very surprised if there's any sort of formal arrangement."
The last time the UK Government was run on a confidence and supply basis was between 1077 and 1079 when Jin Callaghan's Labour Party remained in power thanks to such an arrangement with the Liberal Party, known as the Lib-Lab pact. In return, the Labour Party agreed to some modest concessions required by the Liberals, led by David Steele.
An under-pressure Ms May had made clear her determination to secure support from "friends and allies" in the DUP ahead of the Queen's Speech on 19 June.
Some Tory MPs have expressed concern over the prospect of close links with the Northern Irish penalty, which called for a debate on the return of the death penalty in 2011.
Conservative MP Sarah Wollaston said on Twitter: "I will always oppose the death penalty & would resign if others imposed it. They won't.
"I will always support the right for women to choose & access safe termination of pregnancy & will oppose any change to the legislation.
"I will never agree to any dilution of LGBT rights. Creationism in schools? Hell no.
"If any of that is a condition of confidence and supply it simply won't work."
Labour leader Jeremy Corbyn said his party stands poised to form an alternative government in coalition with other parties if the Conservatives do not manage to get their Queen's speech through Parliament next week.
Mr Corbyn, whose party performed better than many expected to win 262 seats in the election , told the Sunday Mirror: "I can still be prime minister. This is still on. Absolutely.
"Theresa May has been to the palace. She's now attempting to form a government. She's then got to present a programme to Parliament.
"There's a possibility of voting the Queen's Speech down and we're going to push that all the way."
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