DUP leader Arlene Foster said on Tuesday that an agreement between the two sides would be concluded "soon".
May lost her parliamentary majority in last week's election and needs the support of the DUP's 10 lawmakers to pass key legislation.
The British Prime Minister should consult with the opposition Labour Party and others on her Brexit strategy, David Cameron, May's predecessor, says, according to the Financial Times.
But while the DUP is in no rush, the clock is ticking for Theresa May - failure to reach a deal ahead of next week's Queen's Speech could lead to the bill being rejected by MPs.
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The deal also prompted Irish premier Enda Kenny to warn that such an alliance could upset Northern Ireland's fragile peace.
"What we're doing in relation to the talks that we're holding, the productive talks we're holding with the Democratic Unionist Party, is ensuring that it is possible to, with their support, give the stability to the United Kingdom government that I think is necessary at this time", May told a news conference in Paris following a meeting with French President Emmanuel Macron.
The DUP is expected to make a better economic deal for Northern Ireland the central factor for any deal, officially referred to as a "confidence and supply deal".
Conservative former Prime Minister Sir John, who was crucial in bringing peace to Northern Ireland, raised concerns about the impact of a Tory deal with the DUP.
May, who ahead of the June referendum supported remaining in the European Union, has promised to start the Brexit talks next week but opponents of a sharp break with the European Union took her woes as a chance to push back against her strategy.
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The NDP and Greens agreed to combine their votes in confidence motions in the legislature in a bid to topple Clark's government.
The DUP leader is nearly certain to ask for greater investment in Northern Ireland as the price of a deal.
New Environment Secretary Michael Gove has insisted he backs the Prime Minister despite the election result, saying she has "amazing gifts and incredible talents".
The PM told the backbench 1922 Committee on Monday a deal with the DUP would not affect power-sharing talks in Northern Ireland or LGBT rights.
She also accepted in the meeting that her authority was "extremely fractured" and she was no longer "calling the shots".
"Going overseas and being seen to be the prime minister and talking to the president of France.is a classic move to shore up authority at home", said Colin Talbot, professor of government at the University of Manchester.
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"I will have a problem with it if money which is being given to Northern Ireland for purposes that are devolved to Wales and Scotland", he told BBC Radio Wales' Good Evening Wales programme. She took the responsibility for not getting a clear victory and also ensured that she would fix it.