Meanwhile, the prime minister reportedly told her MPs: 'I'm the person who got us into this mess and I'm the one who will get us out of it'.
With all the controversy surrounding the possible deal, after May herself once advocated working to drop the "nasty party" image she said the Conservatives had long attracted, the question remains of whether it will actually come to fruition and, if it does, whether it will last.
Mrs Foster has also rejected suggestions that the mooted agreement could undermine a return to power-sharing arrangements at Stormont amid claims from political rivals that the Government's stated impartiality as a mediator would be fatally undermined.
According to sources, Tuesday's discussions are likely to focus on the government's legislative plans for the year ahead as well as the upcoming Brexit negotiations due to begin on June 19, and their particular implications for Northern Ireland's border with the Republic of Ireland and trade.
"If we are able to do a deal that brings more economic prosperity to Northern Ireland surely that's a good thing for all of our people in Northern Ireland", the DUP leader commented.
It is expected that Conservative manifesto commitments that might have proved problematic for the DUP - such as cuts to winter fuel payments and the removal of the triple lock guarantee for pensioners - will be watered down or ditched altogether.
However, on top of being unpopular with large swathes of the United Kingdom electorate - with a petition against such a deal on change.org receiving over 700,000 signatures as of June 12 - the DUP's historically anti-gay, anti-Catholic and anti-abortion agenda has spooked even Conservative Members of Parliament.
May is under pressure to take on a more cross-party approach to Brexit talks.
The disastrous result also prompted calls - from within her own party - for her resignation, leading May to apologise to her own MPs on Monday evening.
Michael Gove, a former leadership rival, who was appointed environment secretary after losing his post as justice secretary in May's cabinet reshuffle previous year, said the PM is the ideal candidate to lead the country through Brexit. "The union as I've said before is our guiding star", she said. After all, Britain's queen gave her country's prime minister permission to form a government on Friday day last week.
Mr Gove said there was a need to ensure public spending was sustainable but stressed that "we also need to take account of legitimate public concerns about ensuring that we properly fund public services in the future". Foreign minister Boris Johnson, who was reported by British media to be lining up a leadership bid, insisted May should stay.