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.
If May now sends a team to Brussels - ideally on schedule on Monday - that shows willing on the EU's priority "Phase One" issues, then trade talks could get under way by the turn of the year - a step-by-step timetable Barnier says must be followed to limit the risk of a disruptive "no deal".
This would mean the country would still have access to the single market, while being able to make deals without the rest of the EU. "It's our dominating theme today and will remain so over the weekend as bodies are recovered and we start to get a sense of the scale of the tragedy".
After facing criticism for not meeting survivors on Thursday, May visited a hospital on Friday to speak to some of the people who escaped the fire.
The Democratic Unionist Party wants to attach its own conditions onto Brexit as part of its deal with the Conservatives - including the insistence on a "soft border" between Northern Ireland and the Irish Republic.
David Davis is in charge of heading up negotiations as Britain leaves the EU How would it be different to a hard Brexit?
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Nevertheless, it illustrated the challenge May will face in the remaining days before the European Union divorce talks begin: finding a position that satisfies both pro-European and eurosceptic factions of her party if she wants to remain in power. Summit chair Donald Tusk and the other leaders will not respond before May leaves the room to let the 27 discuss their own Brexit strategies."It's absolutely certain that no one will engage with her", a second European Union diplomat told Reuters.
The Prime Minister has insisted the United Kingdom will leave the single market, replace customs union membership with a new deal and end the jurisdiction of the European Court of Justice.
The fact 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 big clash that we feared isn't there", said one senior official involved.
Britain's Brexit ministry said on Friday that no deal could be struck on exiting unless the future relationship with the bloc was taken into account.
According to what an EU official familiar with the negotiations arrangements told New Europe, the EU side is looking at 4-week negotiation cycles, while the initial talks are expected to focus on early issues about the negotiating regime, or as the EU official referred to, it is going to be "talks about talks". "But the withdrawal and future are intimately linked".
Two months ago, the Prime Minister could have looked forward to Britain having a strong negotiating hand on Day One of Brexit talks; now it is hard to see how her position could be any weaker.