May said after a brief meeting with the queen on Friday that the Conservatives will form a new minority government with the backing of 10 MPs from Northern Ireland's Democratic Unionist Party (DUP).
Since the surprise election result on Thursday, the British Prime Minister has insisted that she will remain in post, despite her party losing its majority in the House of Commons.
The agreement stops short of a full coalition after several Tory MPs expressed concerns about the DUP's stance on social issues such as gay rights and abortion. "We have more in common with them than we do the other parties".
Meanwhile, Labour leader Jeremy Corbyn, fresh from his party's strong showing, continues to put pressure on the weakened Conservatives.
That result has obliged a thrashed Tory party, now without an overall majority, to form a working relationship - effectively an alliance - with the Democratic Unionist Party or DUP.
Another Conservative MP, Tom Tugendhat, said: "I joined a party that introduced equal marriage, backs civil rights and defends freedom of faith".
They say she faces the choice between either agreeing to go quietly or facing an immediate leadership challenge, which will throw the party into an internecine war between those who want a sharp rupture with the European Union and those championing a so-called softer Brexit that would see Britain remaining inside Europe's Single Market and accepting continued European migrants.
"From hubris to humiliation", said the left-leaning Guardian. "This is still on", Corbyn told the Sunday Mirror newspaper. The prime minister said that he would bring "considerable experience" to the post.
Defense Secretary Michael Fallon said on the Andrew Marr show he believed the Conservative Party and the DUP will be able to hold together in government.
The Conservative leader has been warned that her days are numbered after calling Thursday's vote three years early, only to lose her majority in parliament. Foster is due to travel to London on Tuesday to meet May to discuss the details of a possible arrangement.
Fallon told the BBC that in light of the election result a new approach was needed, welcoming the resignation of her two closest aides Nick Timothy and Fiona Hill, who were perceived by many Conservative lawmakers to be high-handed and secretive.
Some of Theresa May's most senior ministers are plotting to soften her plans for a hard Brexit, even suggesting the United Kingdom could stay in Europe's single market and customs union, as the prime minister fights to stay in power. The additional 10 votes gets her there.
"I sought, and to be fair to the prime minister, received a categoric assurance that in talking to the DUP that there would be no suggestion of any rollback on LGBTI rights in the rest of the United Kingdom", she told reporters.
May called the early election when her party was comfortably ahead in the polls, in the hope of increasing her majority and strengthening Britain's hand in exit talks with the EU.
The uncertainty prompted Irish republic premier Enda Kenny to phone May Sunday, warning that "nothing should happen" to put at risk divided Ireland's 1998 Good Friday peace agreement, according to his office.
"I could not care less what people get up to in terms of their sexuality". It would include accepting unlimited European immigration, the jurisdiction of the European Court of Justice, and the requirement to pay billions of pounds into the EU budget every year - all of which May promised voters she'd end.Pro-Brexit Conservative lawmaker Bernard Jenkin attacked his party colleagues who want to use May's weakness to dilute her vision of a clean break.
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