"We will welcome any such deal being agreed, as it will provide the stability and certainty the whole country requires as we embark on Brexit and beyond".
Ruth Davidson, the Conservative leader in Scotland, said she had asked May for assurances that there would be no attack on gay rights after a deal with the DUP.
Downing Street initially said an outline agreement on a "confidence and supply" arrangement had been reached with the DUP which will be put to the Cabinet for discussion on Monday.
The DUP has proved hugely controversial in the past over the homophobic and sectarian views of some of its representatives.
This is a measure of May's new fragility since losing her Parliamentary majority, a fragility that will have a huge impact on the Prime Minister's ability to force through a Brexit deal in her own image.
Sunday's newspapers were unsparing, with The Observer writing: "Discredited, humiliated, diminished".
"I was fairly straightforward with her and I told her that there were a number of things that count to me more than the party", Davidson told the BBC.
Sunday papers reported that Boris Johnson was either being encouraged to make a leadership bid in an effort to oust Theresa May, or actually preparing one - a claim dismissed as "tripe" by the Foreign Secretary.
Nicky Morgan, another Tory who was sacked by Mrs May, said the Government must "build a consensus across the Commons" - adding that the PM should "compromise" on the Brexit plans she has already laid out.
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May was under pressure to fire her most senior advisers, Nick Timothy and Fiona Hill, blamed by many Conservatives for steering the election campaign toward disaster.
The Downing St. resignations came as May worked to fill jobs in her minority government and replace ministers who lost their seats on Thursday.
Timothy - a combative character who one former colleague said had helped create a "toxic" atmosphere at the heart of the government - said he took responsibility for the Conservative manifesto, including a plan for elderly social care that caused a backlash.
Graham Brady, who chairs the influential 1922 committee of Conservative members, said there was no public appetite for a second election after May's party failed to win a majority in Thursday's vote.
She seems secure for the immediate future, because senior Conservatives don't want to plunge the party into a damaging leadership contest.
Labour MP Jo Stevens said the prospect of Mrs May's party doing a deal with the DUP was "chilling", after Tory former cabinet minister Owen Paterson said his party may have to enter "a debate I suppose on further reduction of abortion times as medical science advances". One organiser led chants of "racist, sexist, anti-gay, the DUP has got to go".
The outcome risks upsetting the political balance in Northern Ireland by aligning London more closely with the pro-British side in the divided province, and increases the chance that Britain will fall out of the European Union in 2019 without a deal.
"I didn't know who the DUP were, I had to Google them, as many people no doubt in this country would have had to Google them".
Her real test is likely to come when MPs vote on her programme after it is outlined in parliament by Queen Elizabeth II on June 19. But Britain's Saturday newspapers agreed she is just clinging on.
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