Less than six months before Britain is due to leave, the impasse has increased the possibility of a "no deal" Brexit that could potentially disrupt trade, delay movement of goods and starve the world's fifth largest economy of investment.
Mrs May then travels to Brussels on Wednesday to address a dinner for the other 27 European Union leaders.
On Tuesday, officials echoed the mantra that "the clock is ticking" to agree a treaty that parliaments can ratify in time for Brexit.
She was speaking the day after her Brexit minister Dominic Raab visited Brussels but failed to make a breakthrough as hoped ahead of a crucial summit of European Union leaders on Wednesday and Thursday.
The Irish border backstop has emerged as the major sticking point in the Brexit negotiations.
Both sides are now focusing on another summit in November, with the hope that differences can be squared at a further meeting before Christmas.
Labour MP Ian Murray said: "I'm pleased that at Labour conference, party policy changed to support a People's Vote, but it's clear that was achieved despite a lack of support from Jeremy Corbyn".
"We are not there yet", said EU Brexit chief negotiator Michel Barnier in Luxembourg.
Over the weekend the former Brexit minister David Davis called for rebellion, demanding ministers "exert their collective authority" at this "moment of truth".
He also said the fact that the European Union is preparing for a no-deal scenario "must not, under any circumstances, lead us away from making every effort to reach the best agreement possible" for all involved.
Speaking on the eve of the summit in Luxembourg, Barnier said many terms of the divorce have been agreed upon already, such as how much money Britain owes the EU.
A backstop aims to prevent a hard border, including customs posts along the frontier, in the event of the United Kingdom leaving the European Union without a formal exit deal. She says that is unacceptable because it would impose new controls between Northern Ireland and the rest of the UK.
"However, for a breakthrough to take place, besides goodwill we need new facts".
To illustrate the perceived intractability of the Irish border problem, Tusk compared it to the mythical Gordian knot that can not be untied because it is so complicated. He described it as a "new version of the Gordian knot", adding that he couldn't see a new version of Alexander the Great around to solve it.
He said: "I think that is very risky for us and it's something that we have to resist".
Since the Brexit discussions began over 18 months ago, the October summit was earmarked as the most likely date for an agreement given the need for parliamentary approvals before Britain officially departs in March.
There is now growing concern that Britain could crash out of the bloc without any deal at all in March, risking legal and economic chaos. Tusk warned that chances of Britain leaving without a deal were higher than ever before.
During a three-hour Cabinet meeting today, May said the Government must "stand together and stand firm".
But the threat she faces was revealed by Sammy Wilson, the Brexit spokesman for the Democratic Unionist Party (DUP) which props up her government.