Following talks in Downing Street with new Irish Prime Minister Leo Varadkar, Mrs May said the terms of any arrangement between the Conservatives and the DUP would be made public once they were agreed.
The Northern Irish party in talks to support Theresa May's minority government hopes to conclude negotiations with the prime minister's Conservative Party as quickly as possible, its leader told BBC on Friday.
Addressing the Communist Party of Britain's executive committee, he declared that the DUP priority would be to gain more public money and reduce corporation tax on business profits to serve its own narrow interests in northern Ireland.
"They can't have it both ways, it has to be dealt with sensibly", she said.
Speaking before she entered talks with the new Fine Geal Leader, Foster threw down the gauntlet to Sinn Fein and said her party were "ready to dance" in order to restore a power sharing executive up North.
"The most right-wing and reactionary party in northern Ireland will be backed by the most right-wing and reactionary major party in Britain, directly the product of British imperialism's history of intervention and domination in Ireland", he commented, pointing out that the UDA and UVF paramilitaries had backed the DUP in this month's General Election.More news: Man Utd target Morata gets married in Venice
"It continues and when the deal is done, it will be done", a senior source in the Conservative Party said yesterday.
Northern Ireland has been without a powersharing executive since March and without a first and deputy first minister since January, after Sinn Fein collapsed the administration amid faltering trust and relations with the DUP.
Mr Coveney, who was taking part in the latest talks initiative in Belfast for the first time since his appointment last week, said he believed all five parties were up for making a deal.
"There have been positive engagements today between ourselves and Sinn Féin", he said.
"We made the case to her that we would oppose any deal that undermined the Good Friday Agreement", he said.
He said the two prime ministers had spoken about the two governments' "special role" as co-guarantors of Northern Ireland's peace agreement.
"Clearly the DUP are on the wrong side of the argument, cosying up alongside the Tory Government who are disrespecting the mandate of the people here, who asked to remain within the European Union", she added.