Although the DUP are unionists - wanting to remain part of the United Kingdom - and broadly support numerous policies of the Conservative and Unionist Party (as the Conservatives are correctly called), May's proposed deal could scupper attempt to broker a power-sharing deal with Sinn Fein - which promotes the unification of Ireland - on the Northern Ireland Executive.
Foster and her colleagues were expected to ask for concessions on several Tory policies, including the scrapping of the triple-lock pension scheme, but May has said the NI party will have no veto on major policies.
The talks with the DUP follow May's apology to Conservative rank-and-file lawmakers in a meeting Monday that signaled she would be more open to consultation, particularly with business leaders demanding answers about the details on Britain's departure from the European Union.
British Prime Minister Theresa May neared a deal with a Northern Irish Protestant party to save her premiership on Tuesday but faced a tug of war over her Brexit strategy just days before embarking on formal divorce talks with the European Union.
It's understood that Mrs Foster won't be returning to Belfast tonight as planned.
"There is a unity of objective among people in the United Kingdom", May said following a meeting with French President Emmanuel Macron in Paris.
An initial round of talks between May and DUP leader Arlene Foster ended with no agreement on Tuesday, although both sides said they were hopeful of a deal.
"My main concern is the peace process", he said.
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Her failure to win a majority has put May under pressure over her Brexit plans from inside and outside her party and has prompted complaints about her choice of partner due to the DUP's stance on social issues such as gay marriage.
But the prospect of a deal has caused consternation in Dublin, with Ireland's outgoing premier Enda Kenny warning that such an alliance could upset Northern Ireland's fragile peace.
With her tally of Conservative MPs slashed to 317 in last week's poll, Mrs May needs the backing of the DUP's 10 members to reach the 320 required for a working majority in the Commons.
Talks with the DUP to secure her government broke up on Tuesday night without an agreement, but Mrs May said the discussions had been "productive".
Sinn Fein, which won seven seats in the British parliament last week but does not take up its seats or vote in Westminster, would likely reject a deal to form a government by refusing to work with the DUP in Northern Ireland.
Labour's unexpectedly strong second-place showing has thrown national politics into disarray.
But then May called a needless election to get a bigger majority in parliament-to "strengthen her hand" in the negotiations with the European Union that are scheduled to begin next Monday, or so she said. "It's passing quicker than anyone believes That's why we're ready to start very quickly". "I can't negotiate with myself".
France's Macron said the EU's door was still open for Britain as long as the negotiations were not finished, but that it would be hard to reverse course.