May vowed to stay on despite the poor results, and on Sunday unveiled a largely unchanged new cabinet, which met for the first time on Monday.
The prime minister's most prominent potential rival, Foreign Secretary Boris Johnson, sought to quash any suggestion that she would be ousted imminently.
"The old adage, "Where there's a will, there's a way" still holds true, but the majority of people in Northern Ireland are still bluntly asking the Stormont politicians: 'Do you really have the will to make it work?'"
UK Prime Minister Theresa May, clinging on in Downing Street, will appeal for the support of her members of parliament (MPs) Monday, after losing the Conservatives a cherished government majority in last week's election.
It comes after Mrs May told Tory MPs: "I'm the person who got us into this mess and I'm the one who will get us out of it".
And he claimed Britain would still be prepared to walk away from negotiations without a deal if necessary.
"The reason for leaving the single market is because we want to take back control of our borders - they are not compatible".
Outgoing Irish Prime Minister Enda Kenny voiced concern in a phone call with May that the proposed deal between her Conservatives and the DUP could undermine the Northern Ireland peace process.
May had been hoping that a huge majority would allow her to ditch ministers who are seen as a liability, or against whom she entertains animosity.
Senior Cabinet minister offer support to UK prime minister
May tried to reassert her shattered authority at the weekend by announcing her new cabinet - with no changes among her top team. DUP is strongly anti-abortion and anti-same sex marriage, for example - stances that aren't very popular in the rest of the UK.
Theresa May's immediate reaction to the last Thursday's general election results was reportedly one of stunned silence, before the Prime Minister "burst into tears".
May has a busy schedule on Tuesday, hosting a cabinet meeting and talks with the DUP leader before travelling to Paris to meet French President Emmanuel Macron.
"On valuation grounds, sterling is about fair value versus the other major currencies, so to see a much cheaper currency would require considerable new news on either the state of the United Kingdom economy or on the state of Brexit negotiations".
"If the DUP don't prioritise the restoration of the institutions, and instead decide to become a prop for a dysfunctional minority government in London, then the parties should consider inviting an independent chairperson to oversee proceedings".
May tried to reassert her shattered authority at the weekend by announcing her new cabinet - with no changes among her top team.
Commenting on Gauke's appointment, Royal London director of policy, and ex-pension minister, Steve Webb, said: 'There are few ministers who could have been appointed to this role who know as much about pensions as David Gauke. The pro-Brexit Gove was appointed Environment Secretary.
"However, the Conservative Party's reduced share of the vote may indicate a higher likelihood that a "softer" form of Brexit might now be pursued, involving compromises with the European Union that Ms".
DUP leader Arlene Foster said there had been "positive engagement" so far.
He added he was going to do what he thought was in the national interest, and that was to keep the Prime Minister in office.
"I have no doubt over time those responsible will look foolish in the extreme", she said.
"In truth, no-one expected the outcome of the snap general election to be a hung Parliament, and for the DUP to be in such an influential position", she said. While British journalists and voters frantically googled "what is the DUP", those of us living in England spent the day educating the people around us on the policies and positions of the Conservative Party's new best friends.