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 EU says it wants to avoid the return of a "hard border" with Ireland that would require passport controls and customs checks, but how that will be possible without Britain staying part of the EU single market or customs union is not clear. If she manages to remain in Downing St at all, it will be with a Cabinet demanding more collegial decisions.
But after a generation of discord over Europe inside her party, May's future could depend on her ability to please both the eurosceptic and pro-European factions in her party. "So I don't know whether we would do that, but we might say, "OK, we will not provide health services free of charge" - that might be a Spanish position, for example".
The newspaper quoted another former minister as saying: "If she weakened on Brexit, the world would fall in. all hell would break loose".
After a tumultuous week that pitched Britain into its deepest political crisis since the Brexit referendum a year ago, May's future was uncertain due to her botched gamble on a snap election and a muted response to a deadly fire in London.
Following talks in Dublin with the new Irish Prime Minister Leo Varadkar, she said that reaching a "sensible" Brexit had been the focus of their talks.
The Conservatives now only have 317 seats in the 650-seat House of Commons and need the support of the DUP's 10 MPs for a razor-thin majority.
The Queen's speech was due to be delivered on Monday but has been delayed to give the Tory minority government more time to strike a deal with the Democratic Unionist party.
Finance minister Philip Hammond has led calls for a softer strategy on Brexit, prioritising the economy.
The third key issue is the future of the peace process in the British province of Northern Ireland, and the status of the border with the Republic of Ireland.
The fact Mr Davis agreed to an agenda on Monday that will see him and British officials spend some seven hours in the European Union executive's headquarters discussing who will talk to whom when and about what has come as a relief to some European Union negotiators.
The Commission, which is leading Brexit negotiations on behalf of the European Union, insisted the process would start in line with its "sequenced approach" to negotiating the UK's departure.
Many in Britain have seen the election result as repudiating May's threats to walk away without a deal. "Everything must go through Barnier", said a third, underlining European Union strategy to keep talks technical for now.
Thus whether Brexit can be halted or not depends primarily on the UK's government and people.
The agenda for the opening day of talks begins at 11 a.m. with an introductory session involving Barnier, the U.K.'s Brexit Secretary David Davis and officials.