After attending the Prime Minister's Cabinet meeting on Monday, Ms Davidson told BBC News: "I'm suggesting that the Conservative Party works with those both within the House of Commons and with people without to ensure that as we leave the European Union, we have a Brexit that works for the economy and puts that first".
The agency, quoting two unnamed sources from the meeting, said May told lawmakers that she was the one "who got us into this mess" and that she was the "one who will get us out of it".
The Conservatives won 318 seats out of the 650 in last Thursday's vote, meaning they now require support to govern and have chosen the eurosceptic Northern Irish Democratic Unionist Party (DUP), which has 10 seats, for that.
Mr Green said the Brexit talks were "the most urgent task facing the new Government".
Government spokesman Steffen Seibert says Germany will apply to move the medicines regulator to the western city of Bonn, which already hosts several national and global agencies. May should be able to use these two set-piece events to shape the country's future.
Talks between the DUP and Irish nationalists Sinn Fein on forming a power-sharing government are due to restart in Belfast later on Monday.
The prime minister's official spokesman insisted on Monday that nothing has changed in the government's approach to Brexit.
The Daily Telegraph understands that more than a dozen Tory MPs have significant concerns about the prospect of the deal and have warned it could lead to the collapse of the Good Friday Agreement, the political milestone in the Northern Ireland peace process of the 1990s.
But even in this her lack of authority prevented her from major moves or sackings.
The Sinn Féin leader confirmed the party's "magnificent seven" new MPs would not take up seats in Westminster.
Mr Davis added: "In the first round we are going to have pretty long meetings at roughly one week a month - which is much, much faster than any previous trade deal they have done. This doesn't depend entirely on us".
UK Prime Minister Theresa May, clinging on in Downing Street, will appeal for the support of her members of parliament (MPs) Monday, after losing the Conservatives a cherished government majority in last week's election. Already there are expectations that the Queen's Speech, which would outline the government's legislative agenda for next year, will be delayed from next Monday. Article 50 was triggered in March, giving the United Kingdom two years to negotiate the terms of its divorce from the 28-state block.
The European Commission said it is prepared for the negotiations to start as soon as the United Kingdom is ready but refused to give a date.
Brussels has warned that time is running out to start the talks on divorce terms and a future trade deal, with Britain set to leave the European Union come what may in March 2019.
For the British Government to now create a political alliance with one of those parties clearly poses a conflict of interest, at the very least, and, at its worst, could derail the progress made through the Good Friday Agreement and the establishment of the Northern Ireland Assembly entirely.
The Prime Minister would serve the people of Britain - and Northern Ireland in particular - far better now by listening to them and acting on the messages they have sent her through the election result and all that has followed since.
"I have no doubt over time those responsible will look foolish in the extreme", she said.
Davis told ITV he's loyal to May and there's a distinction between "running a campaign and running a country".
As reported in The Guardian, Theresa May has singlehandedly weakened her position, and many in her own party have declared that her position is "untenable".
A Tory-DUP pact is also a partnership that brings into mainstream British politics a party which is opposed to societal ideals widely held by the British public - such as rights for the LGBT community and on abortion and environmental values around climate change.