DUP leader Arlene Foster, who was accompanied by deputy Nigel Dodds, said she believed there would be a "successful conclusion" soon.
The Irish republican Sinn Fein party - which won seven seats in the election although their MPs traditionally do not take up their seats in protest - is also wary of the alliance. In Dublin, Mrs Foster brushed aside Sinn Féin assertions that DUP support for the British Government compromises the 1998 Good Friday Agreement.
Smoke billows from a tower block severly damaged by a serious fire, in north Kensington, West London.
The 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.
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.
May said on Tuesday that talks with the DUP had been productive - a view shared by DUP leader Arlene Foster - and that Brexit negotiations would begin as planned next week.
The proposed deal would see the DUP back the Conservatives in votes on the Budget and confidence motions.
"I want to record my deep-seated concern and, I believe, the deep-seated concern of many, not just in Scotland, but across the United Kingdom right now, at the prospect of some sort of grubby deal between the Tories and the DUP to allow Theresa May to cling to office", she said.
Former prime minister David Cameron said May needed to listen to opposition parties, and that there would be pressure for a softer Brexit that would give greater priority to a close trading deal with the EU.
"It's going to be hard".
Major said: "We need to be prepared for the unexpected, we need to hope for the best but prepare for the worst".
The Times newspaper said finance minister Philip Hammond would push May not to leave the customs union - an arrangement that guarantees tariff-free trade within the bloc but prohibits members from striking trade deals with third parties.
That would help safeguard jobs and trade with European Union members, but would severely restrict the UK's ability to strike its own trade deals around the world.
French President Emmanuel Macron (R) and Britain's Prime Minister Theresa May attend a joint press conference at the Elysee Palace in Paris, France, June 13, 2017.
"I want us to have a good working relationship so that from his jurisdiction and my jurisdiction we can work together for the good of all our people", she added.
Mrs May stressed Brexit would happen and the timetable remains on course.
"I congratulate her on returning and I'm sure she'll agree with me that democracy is a wondrous thing, and can throw up some very unexpected results", he said. "The intent is to ensure that we have the stability of Government in the national interest".