May has dismissed calls to resign following the dismal election result after calling a vote three years early in the hope of bolstering her slim majority, only to actually lose seats. Without the DUP's support, May can not be sure of doing so.
European Union officials see the start of Brexit talks on Monday as a sign Theresa May is accepting their format for negotiations but they expect no quick deals and are wary the prime minister may try to break with Brussels protocol.
Sources also told the BBC that both parties were close to reaching an agreement so that May could form the government but the fire at Grenfell Towers in London made an announcement on the deal on Wednesday "inappropriate".
Davis plans to go to Brussels on Monday to start the negotiations, which will reshape not only Britain's role in the world, but also that of a bloc praised for ensuring peace after World War Two.
Foreign Minister Sigmar Gabriel, echoing remarks by French President Emmanuel Macron, said the door to Europe remained open to Britain, and it was up to the British people whether they wanted to change their minds about leaving the EU.
"And we itemised those matters in which she was dilatory or in default in relation to that agreement", he added. "We believe that the withdrawal process can not be concluded without the future relationship also being taken into account".
A lackluster campaign saw her high approval rating slip away, and support for her "hard Brexit" strategy - pulling out of the European single market and customs union - now hangs in the balance.
"Look at what the Tories (Conservatives) have managed to do to the United Kingdom in the space of just one year, firstly calling a divisive and reckless European Union referendum. then having lost that gamble pursuing a hard Brexit path", Scotland's first minister, Nicola Sturgeon, said.
May's finance minister, Philip Hammond, said on Friday that Britain should prioritize protecting jobs and economic growth.
"The UK Government is offering whatever support we can, working alongside the Irish government, as appropriate, honouring our respective commitments in the Belfast Agreement to serve the interests of the whole community in Northern Ireland", he said.
The EU's executive Commission said in a statement Friday that the first round of negotiations in Brussels will be part of a "sequenced approach to the talks".
While Monday's session is billed as "the opening of negotiations at the political level" - with citizens' rights, the financial settlement, the Northern Irish border issue and more due to be discussed - in reality the meeting will mostly be a more formalized continuation of the "talks about talks" that have been underway for weeks.
"It will take us several months to draw out the conditions of an orderly withdrawal... so let's not waste time", he said.