Nigel Dodds has spoken in the House of Commons.
The talks with the DUP follow her apology to Conservative rank-and-file lawmakers in a meeting for the party's poor election result.
Weakened by an election that was meant to strengthen her, May is now working to regain some of her lost standing.
Addressing a meeting of backbenchers, the PM reportedly said she would serve as "long as you want me to do".
Mrs Foster said: "We've had some very good discussions today and those discussions are continuing this afternoon".
Supporters of "hard" and "soft" Brexit tried to take advantage of the political chaos in Britain on Monday to promote their visions amid fears that their rivalry could revive old divisions in the Conservative party.
During the campaign, May cast herself as the only leader competent enough to navigate the tortuous Brexit negotiations that will shape the future of the United Kingdom and its $2.5 trillion economy.
"Coalition of chaos" was the slogan the Tories had used to describe a Labour minority win during the election campaign.
Britain's Brexit minister, David Davis, has backed May, saying claims made by former finance minister George Osborne that she is a "dead woman walking" are wrong and self-indulgent. "This isn't just going to be a Tory Brexit, this is going to have to involve the whole country".
"We need to get on with the job of reducing the deficit so that we do not saddle the next generation with the burden of debt, and the larger the deficit the more money that should be spent on health and education is actually spent on paying down debt". He paid tribute to the Northern Ireland MPs who were not re-elected.
DUP leader Arlene Foster is due to meet May on Wednesday in order to thrash out the details of that deal, which will likely entail a series of concessions in exchange for DUP support in the Commons.
It is thought Mrs Foster, despite being a Brexit supporter, could seek assurances from Mrs May that she will pursue a softer exit from the European Union, given Northern Ireland's 56% Remain vote and the DUP's desire not to see a return to a hard border with Ireland.
He also said that there should be "some agreement on the immigration that we can accept from Europe".
"The UK has had a reputation, earned over the generations, for stability and predictability in its government", said a senior executive at a multi-national company listed on the London FTSE 100, speaking on condition of anonymity.
DUP leader Foster said on Twitter: 'Discussions are going well with the government and we hope soon to be able to bring this work to a successful conclusion'.
Johnson, meanwhile, has brushed off reports he is plotting to oust May, insisting he fully supports her attempts to form a minority government.
The party will offer support for key votes like backing the Queen's Speech, the budget and a vote of no confidence.
Scottish Conservatives sharply increased their representation in the Westminster parliament last week, in contrast to the party's losses in England, strengthening the influence of their leader Ruth Davidson within the party.
"But it remains another question if I'll still witness this", added the 74-year-old.Asked about Schaeuble's comments, 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.
It has run successfully - if under some strain because of historical tensions between the two parties - until the resignation of McGuinness, January 9, 2017, over a row between the two parties over a botched green energy scheme originally overseen by Foster, which was supposed to incentivize people to switch to renewable, but which ended in farce when the scheme paid out more than the cost of the fuel, leaving taxpayers with a bill of up to US$486 million.