She is seeking a so-called confidence and supply deal, which would involve the DUP supporting the Conservatives on key votes but not joining a formal coalition.
While May's top team has been left unchanged, she will have to fill gaps in her ministerial team after nine junior ministers lost their seats in what has been characterised as a disastrous election night for the ruling party, with the shock results going against every pre-election opinion poll forecast and the Jeremy Corbyn-led Opposition Labour faring far better than predicted.
Mrs May failed to win a majority in the British parliament in an election on Thursday.
Labour leader Jeremy Corbyn said he could still be prime minister, although his party has no obvious way to build a majority coalition.
The British PM was forced to relinquish her two closest aides - joint chiefs of staff Nick Timothy and Fiona Hill - as she struggled to reassert her authority. Their influence had increasingly angered senior ministers.
Britain's largely pro-Conservative press questioned whether May could remain in power.
May suffered a humiliating blow as the "snap election" Thursday spectacularly backfired, stripping her Conservative Party of its commanding majority in Parliament.
"President Trump emphasised his commitment to the United States-United Kingdom special relationship and underscored that he looks forward to working with the Prime Minister on shared goals and interests in the years to come", the White House said in a statement on the telephone conversation between the two leaders.
A party spokesman confirmed the resignation of Hill, a combative character who one former colleague said had helped create a "toxic" atmosphere at the heart of the government.
Senior Conservative lawmaker Graham Brady said the prospect of being propped up by the socially conservative DUP, which is strongly focused on Northern Ireland's specific political complexities, was causing concern in his party.
Silva and Diego Costa on target as Spain beat plucky Macedonia
Spain leads Italy on goal difference before the two meet in Madrid on September 2. Ukraine beat Finland 2-1 and Turkey won 4-1 against Kosovo .
The turmoil engulfing May has increased the chance that Britain will fall out of the European Union in 2019 without a deal.
Mrs May's party is deeply divided over what it wants from Brexit, and the election result means British businesses still have no idea what trading rules they can expect in the coming years. "That's not a matter for me", she said.
In an exclusive interview with the Sunday Mirror, Mr Corbyn is champing at the bit and buzzing with enthusiasm.
She also confirmed to German Chancellor Angela Merkel in a phone call yesterday that Britain was ready to begin Brexit negotiations "as planned in the next couple of weeks", reassuring European Union leaders who had expressed doubts after May's electoral losses.
"I think if we'd had a large Conservative majority like all the polling was pointing to, then Theresa May could've pushed through her version of Brexit which is outside the single market, outside the customs union with tight control over migration, and not giving much money into the EU pot in terms of paying for access", explains French.
By contrast, the Labour campaign focussed overwhelmingly on its aspirations to build a nation "For the Many, Not the Few", cleverly shifting the debate away from Brexit to the impact that years of austerity, including for the past seven years under the Conservatives, had had on voters - from the schools their children attended to the pressures facing the National Health Service.
A failure to get legislation through Parliament could eventually trigger another election.
"It is not the outcome any of us would have wanted in the Conservative Party".
"Theresa May is certainly the strongest leader that we have at the moment", David Jones, a junior Brexit minister, told the BBC.
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