Discredited, shattered and weakened after the catastrophic vote, the 60-year-old prime minister is now faced with overwhelming pressure from several prominent Conservatives, and other voices in Westminster and beyond to abandon her hardline stance.
Even May's own immediate political survival is in doubt, 10 days after she lost her majority in an election.
The agenda for the opening day of talks begins at 11 am with an introductory session involving Europe's chief Brexit negotiator, Michel Barnier, and Davis. They are due to give a joint news conference after talks among their teams lasting seven hours.
The terms of reference state "for both parties the default is transparency" and that it is for the side providing any information to state whether it should remain secret.
That the British were ready to break negotiations down on the lines of priorities identified by Brussels implied, he said, a willingness to start a first phase of divorce talks before moving on next year to a second phase on the new free trade relationship May wants - a "sequencing" London has disputed.
While 53 percent of those surveyed outside of the United Kingdom would like votes on whether to remain within the region, only 18 percent said they would back leaving. Mr Barnier, a keen mountaineer, spent the weekend in his native Alps "to draw the strength and energy needed for long walks".
The 60-year-old prime minister emerged victorious from a bitter and divisive leadership battle after her predecessor, David Cameron, quit following Britain's vote a year ago to leave the EU.
"The whole process will lead to a happy resolution - I think it can be done with profit and with honor for both sides", Johnson said on the sidelines of an European Union foreign ministerial meeting in Luxembourg.
"We are. determined to build a strong and special partnership between ourselves, our European allies and friends", Davis said before the talks.
Fighting for her political survival, May has been trying to strike a deal with a small Northern Irish Protestant party to avoid a second election that could delay Brexit talks and damage the $2.5 trillion economy.
Like Green, the pro-business and pragmatic politician prefers a soft Brexit that focuses on jobs, business and economic growth and is now thought to be heading a battle within May's minority government to keep Britain inside the EU's customs union and single market.
He added: "It's not as if Europe is leaving Britain; Britain wants to leave the EU".
Britain faces the hard fact that the community has more pressing matters to discuss than Britain's parting wishes.
"We want to end the anxiety facing 4 million citizens", he said. He added: "Brexit won't make anything better, but it will make a lot of things more hard". But there is also incredulity that the British government is preparing to go into the talks with a hard-line negotiating position when it can not command a majority at home.
He added: "As the European Union has itself said, "nothing is agreed, until everything is agreed". But the turmoil of the euro zone crisis, fears in Britain about immigration and a series of miscalculations by former Prime Minister David Cameron prompted Britain to vote by 52 to 48 percent for Brexit in a June 23 referendum past year. The election of the fervently europhile Macron, and his party's sweep of the French parliament on Sunday, has revived optimism in Brussels.style="text-align: center;"