DUP leader Arlene Foster, who travelled to Westminster for talks with the Tories on Tuesday, said she hoped a deal could be reached "sooner rather than later".
But pressure was mounting for May to change course on the type of Brexit Britain should pursue.
"My preoccupation is that time is passing, it is passing quicker than anyone believes because the subjects we have to deal with are extraordinarily complex", he said, according to the Financial Times. "We're ready to dance".
Though Foster supported Brexit, she also might demand that May pursue a cushioned exit from the European Union, given her party's wish that a soft border remain between Northern Ireland and Ireland, an European Union member.
European Union officials played down the importance of Britain's lack of a clear final plan, saying that talks on other issues can go ahead without deciding yet on a new relationship.
The DUP have a deadline of June 29 to get a deal done with Sinn Fein in order to restore order in Stormont.
May is expected to form the new government this week, before the Brexit negotiations with the European Union start on June 19, the very day the Queen's Speech - a ceremonial event at the opening of the new government where the queen reads out a text on future government policies - was supposed to take place.
Major warned that although Northern Ireland was a long way from returning to the violence that killed 3,600 people, he believed the peace process remained fragile almost two decades after a US -brokered 1998 peace agreement.
The Prime Minister will meet separately with representatives of Sinn Fein, the Ulster Unionists, the SDLP and the Alliance Party - as well as the DUP - in Downing Street in an attempt to allay growing concerns.
"We were very clear that it can't be a deal that gives the DUP power over the Tory Party", he said.
Labour leader Jeremy Corbyn countered with a bit of previously unseen swagger, wearing a huge red rose - his party's symbol - in his lapel as he sparred with May and taunted her about the uncertainty surrounding the upcoming vote on her legislative program, known as the Queen's Speech.
"It's imperative that both governments recommit to the word, spirit and implementation of the Good Friday Agreement if there is to be any prospect of re-establishing the Executive", O'Neill said in a statement.
Mrs Foster has also rejected suggestions that the mooted agreement could undermine a return to powersharing arrangements at Stormont amid claims from political rivals that the Government's stated impartiality as a mediator would be fatally undermined.
"There is an irony to being lectured by some about our role in the national government of the United Kingdom when Sinn Fein want to be in government here in the Republic of Ireland", she said.
Any hard border, he said, would be catastrophic. However, "a little side bargain to keep Theresa May in power, a temporary little arrangement, won't have any integrity".
SDLP leader Colum Eastwood said the Prime Minister had not convinced him the "DUP tail is not wagging the Tory dog".
Negotiations between the Conservatives and the DUP are understood to be focused on measures to boost the economy in Northern Ireland, as well as proposals to drop some unpopular Conservative manifesto proposals.
The nationalist Sinn Fein and SDLP and the cross-community Alliance have all made clear Northern Ireland Secretary James Brokenshire can not chair the ongoing process to restore power-sharing at Stormont due to the perceived conflict of interest.
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