After Mrs May addressed the 1922 Committee of backbench Tory MPs on Monday, some of those present suggested the PM would consult more with business on her approach to the talks.
Foster said her party would go into the talks "with the national interest at heart".
The prospect of such a deal has already raised alarm in Dublin, where Prime Minister Enda Kenny has warned of the impact on peace in the British province, while the DUP's ultra-conservative views have also prompted criticism.
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.
"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.
The crunch talks could also force the delay of the government's presentation of its legislative programme to parliament by Queen Elizabeth II, due on June 19.
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"We have worked as a party with the DUP before and those are productive talks".
While Mrs May appears to have seen off the threat of an immediate leadership challenge, her weakened grip on power has put her under pressure on several fronts.
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The attorney general testified under oath that he "did not have communications with the Russians" during the campaign. Sessions did not say in the letter whether his appearance would be in public or behind closed doors.
Labour leader Jeremy Corbyn turned Mrs May's election slogans against her, claiming a link-up between the Tories and DUP would be a "coalition of chaos".
Her performance was roundly praised by Conservative MPs.
New Environment Secretary Michael Gove has insisted he backs the Prime Minister despite the election result, saying she has "amazing gifts and incredible talents".
May apologised to Conservative MPs on Monday, accepting personal responsibility for failing to win an outright victory and sacrificing the parliamentary majority she inherited from her predecessor David Cameron when she came to power previous year following the UK's vote to exit the European Union (EU). The Evening Standard, edited by ex-Treasury chief George Osborne, reported that cabinet ministers have initiated talks with Labour lawmakers to come up with a "softer", less hard-line divorce from the EU. "I am dubious about it", Major said.
"Being seen to be the prime minister" could help "shore up her authority at home", according to Colin Talbot, professor of government at the University of Manchester.
"She was generous in acknowledging how sad we naturally felt that some of our colleagues had lost their seats and that was as a result of this election campaign being called at the time it was called".
Ms Davidson said: "We do have to make sure that we invite other people in now".
He said: "It was great to have the chance to chat to Boris over the course of the weekend".
"What we are doing in relation to the productive talks that we are holding with the Democratic Unionist Party is ensuring that it is possible to, with their support, give the stability to the UK Government that I think is necessary at this time. We need to hope for the best but be prepared for the worst".
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