Despite the uncertainty over her ability to govern, May had confirmed that Brexit negotiations - expected to be the most complex worldwide talks Britain has held for decades - would begin as planned next week.
In a joint statement issued, officials said: "David Davis, Secretary of State for Exiting the European Union, and Michel Barnier, the European Commission's Chief Negotiator, agreed today to launch Article 50 negotiations on Monday, 19 June".
The Conservatives are yet to agree a deal with the DUP, a week after the general election stripped May of her parliamentary majority.
Former Prime Minister David Cameron said May needed to listen to rival political parties, and that there would be pressure for a softer Brexit that would give greater priority to a close trading deal with the EU.
"If there is the expected positive outcome, it will be at least the start of next week before anything is signed off", a DUP source told Reuters.
"It takes two to tango and we're ready to dance", she said.
The Conservatives have 317 MPs while the DUP have 10.
Talks to restore confidence took a back seat in recent days as the political focus largely shifted to London and the DUP's deal to prop up the Conservatives at Westminster.
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The Conservative source said this meant the party was "confident" it had enough votes for the programme to be approved, after May suffered a disastrous setback in snap elections a week ago that saw her lose her majority in the 650-seat House of Commons, just ahead of crucial Brexit negotiations with Brussels.
"The risk is that Northern Ireland continues to fail to find solutions, and potentially the peace process unwinds", Simon Usherwood, senior politics lecturer at Surrey University, told AFP.
But the DUP leader insisted it was now down to Sinn Féin whether an agreement is done.
While the DUP are deeply eurosceptic, they have balked at some of the practical implications of a so-called hard Brexit - including a potential loss of an open border with the Republic of Ireland - and talks will touch on efforts to minimise the potential damage to Northern Ireland.
But speaking in Dublin after a meeting with the new Taoiseach Leo Varadkar, the DUP leader Arlene Foster said it was down to Sinn Fein whether an agreement is reached.
Mrs Foster said she thought it was "very realistic" that a deal can be reached on power-sharing in Northern Ireland by the end of this month.
"A little side bargain to keep Theresa May in power. won't have any integrity and certainly isn't as important as the needs of people" in Northern Ireland, said Adams.