But Irish nationalists Sinn Fein have voiced concerns a tie-up could destabilize politics in Northern Ireland and undermine the British government's neutrality in overseeing talks to form a new power-sharing government for the province.
DUP leader Arlene Foster, who was asked to resign in the wake of a heating incentive scandal that cost the government hundreds of thousands of pounds, said on Twitter that discussions are "going well" and that there will be an agreement of sorts.
Theresa May needs to come to an arrangement with the DUP in order to be able to get their ten members of parliament to add to her MPs in order to gain a majority in the House of Commons when voting on legislation.
"My understanding is that the DUP, like the Conservative Party, has a shared interest in making sure we have a stable government", he told BBC radio.
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.
"The issues that have to be dealt with have been talked about now for quite a period of time", she said.
The British government has set a deadline of 29 June for the institutions to be back in place, raising the spectre of further Assembly elections or direct rule if devolution is not restored.
The Conservative source said: "We're confident of getting an agreement, we're confident that the Queen's speech will be passed".
"We warned Mrs May that the pact between the Tories and the DUP has the potential to undermine past agreements and the re-establishment of the Executive".
The pair got to know each other when they were each responsible for tourism policy as ministers.
It is not yet clear whether the European Union withdrawal talks will go ahead on that day, although Brexit Secretary David Davis has said they will start "next week".
He said that he believed that if a deal is reached, the government "will not be seen to be impartial" which will create problems for communities in the North.
The Conservative source said the talks to leave the European Union would not be delayed, removing the question mark over the negotiations being derailed by May's lack of a parliamentary majority lost in an election she did not need to call.
Brexit minister David Davis plans to go to Brussels on Monday to start the talks, which will reshape not only Britain's role in the world, but also that of a bloc praised by many for ensuring peace after World War Two.
The deal between the Conservatives and the DUP, which has not been finalized, was also criticized by Scottish First Minister Nicola Sturgeon.