Ms Davidson said: "I was fairly straightforward with her and I told her that there were a number of things that count to me more than party".
"As we are the party that won most seats and most votes, we are the only party that is in a position to form a government that can do that".
Martin Selmayr, senior aide to European Commission President Jean-Claude Juncker, responded to the resignations by tweeting the word "bauernopfer" - German for the sacrifice of a pawn in chess.
The DUP is a socially conservative pro-British Protestant group that opposes abortion and same-sex marriage and once appointed an environment minister who believes human-driven climate change is a myth.
"We can confirm that the Democratic Unionist Party have agreed to the principles of an outline agreement to support the Conservative government", a spokesman for May said.
Downing Street said an outline agreement on a "confidence and supply" arrangement had been reached which will be put to the cabinet for discussion on Monday.
May had planned to negotiate for a so-called "hard Brexit", which would see the United Kingdom leave the EU's single market for failing to allow freedom of movement.
Northern Ireland is the only part of the U.K.in which same-sex marriage is illegal.
The DUP was founded in the 1970s by the late firebrand preacher Ian Paisley, and in the 1980s was a key player in the "Save Ulster from Sodomy" campaign, which unsuccessfully fought against the legalization of gay sex. Former Conservative cabinet minister Owen Paterson, asked about her future, said: "Let's see how it pans out".
But she has been forced to persue a deal with the Democratic Unionist Party to ensure she can still pass legislation through the House of Commons, where now a majority are fundamentally opposed to the majority of Tory policies.
DUP Leader Arlene Foster recently denied the party was homophobic. "When it becomes a matter for me is when people try to redefine marriage".
Cllr O'Neill is now calling for Mark Lancaster and Iain Stewart to follow other Tory MPs and speak out against any such deal and oppose any Tory deal with the Northern Irish party.
Labour leader Jeremy Corbyn, riding a wave of acclaim for his party's unexpectedly strong showing, called on May to resign.
Newspaper headlines saw her as just clinging on.
Senior party figures have cautioned against any immediate leadership challenge, saying it would cause only further disruption as Britain prepares to start the Brexit talks as early as June 19.
However, Mrs May insisted that, as the leader of the largest party in the new parliament, she had a duty to act in the "national interest" and lead the country through the Brexit negotiations starting in ten days' time.
London attacks: What we know
After ramming the auto into a barrier outside the Houses of Parliament, the driver got out and stabbed a police officer to death. European security official says it is too soon to say whether incidents around London Bridge are militant attacks.
IL native Thad Matta out as coach at Ohio State
That means that, as of June 5th, one of college basketball's 15 best jobs is now on the market, which is a shame for Buckeye fans. The speculation surrounding the job in Columbus has already started with numerous names involved in the in search recycled again.
Worst is still to come‚ forecasters warn storm-tossed Cape Town
He said in one incident a family of four died in a fire caused by lightning, while another person died due to a building collapse. On Wednesday, authorities urged residents to capture rain from the storm in containers for flushing toilets and other needs.
"No one would survive Russia-US war", says Putin
To which Putin responded: "I am not a woman, so I don't have bad days". "There are certain natural cycles". In response to question Putin laughed and said , "Well, I'd prefer not to go into the shower with him".
Ferrari has chosen its number one driver - Hamilton
They chose to run him long and hope to gain from his speed once he got some clear air, as cars pulled in to make their pit stops. With the win, Vettel is now 25 points ahead of Hamilton for the lead in the driver's standings .