In 10 days she will face down her restive Conservative party at its annual conference in Birmingham, amid a growing campaign from her disillusioned colleagues for her to ditch her Brexit plans or quit.
"Everybody shared the view that while there are positive elements in the Chequers proposal the suggested framework for economic cooperation will not work", he said.
But EU member Ireland and the rest of the EU have demanded a "backstop" guarantee that if that happens, then Northern Ireland would have a special status - effectively remaining inside the EU economic space, but at the cost of new differences with the British mainland, which both Mrs May and her vital parliamentary allies from the province say threaten British sovereignty.
On Tuesday, the EU's Brexit chief negotiator Michel Barnier said the bloc was ready to improve its proposal for an "insurance policy" backstop arrangement on how to manage the Irish border after Brexit.
May promised to bring forward new proposals that would ensure trade could move freely across the Irish border, but would not require different customs regimes in Northern Ireland and the rest of the United Kingdom, a key red line for the British Prime Minister.
The Prime Minister's minority government allies the DUP also pressed her to demonstrate a determination "not to be bullied" by the European Union in what is shaping up to be a tense final month of negotiations as time runs out to reach a deal.
A "no deal" Brexit, bringing uncertainty and insecurity for expatriates, has never looked more likely, however the EU27 leaders have agreed to push back the deadline for a withdrawal deal to be agreed to a one-off summit in mid-November since it is now clear the previous deadline of October 18 will be missed.
Diplomats say that while for countries such as France and Germany it is important to make the point with regards to Brexit that a country is better in the European Union than out, the governments of Poland and Hungary are less keen to make Brexit seem like a dead end. On July 6, 2018, after a marathon 12-hour showdown at Chequers, May finally managed to secure a Brexit plan to present to the EU.
Earlier EU leaders called the prime minister's bluff by slamming her post-Brexit plan for trade and refusing to rule out a no-deal scenario.
Think your friends would be interested?
After two-day summit, German Chancellor Angela Merkel said "it was clear today that we need substantial progress by October and that we then aim to finalize everything in November".
However both sides still want a broad-brush description of the relationship, to avoid a so-called "blind Brexit" without a good idea of how trade and borders will work after the transition.
They realise May is the only one who can seal a deal on Brexit and they do not want her to "go belly up", the diplomat said.
She pressed her case for a Brexit deal, telling fellow leaders: "The onus is now on all of us to get this deal done".
The summit dinner was dominated by discussions over how to deal with irregular migration into Europe, but May gave a short speech on Brexit at the end, officials said.
There was no immediate response to his comments from either Downing Street or the Department for Exiting the European Union, and it was unclear whether they would be sufficient to unlock the negotiations. But she tried to downplay their unexpectedly strong criticisms of her Chequers plan. We don't want to get there, we don't expect to get there. "But if we all act responsibly, we can avoid a catastrophe", he said.
"Europe isn't an a la carte menu", French President Emmanuel Macron said, noting that "as it stands, the Chequers plan seems to be a take-it-or-leave-it plan".