She will have to get any deal past her Conservative Party - split between "hard" and "soft" Brexit factions - and past her parliamentary allies in Northern Ireland's Democratic Unionist Party, who insist a solution can't include customs checks between Northern Ireland and the rest of the UK It must also be approved by Britain's Parliament, where May lacks an overall majority.
British Prime Minister Theresa May met with French President Emmanuel Macron in Brussels on Wednesday shortly before addressing fellow European Union leaders on the stalled Brexit negotiations, her office said.
However, the speech, which was followed by a dinner of the 27 leaders that May did not attend, was deemed to contain none of the substantial new proposals that had been called for by European Council President Donald Tusk.
Theresa May is set to attend a Brussels summit on Wednesday to try to move talks forward (WPA Pool/Getty Images)Could it actually go ahead?
But a senior European Union source said after the prime minister's speech: "There were no new proposals".
An EU diplomat told the Financial Times that the possible extension proposal was "an example of how we could be flexible to help the British side if they want it".
The Brexit negotiations may have hit an impasse over how to solve the Irish border issue but there is growing feeling in Brussels that a deal will finally be struck before the March 29 2019 deadline.
The so-called backstop is created to prevent a hard border on the island of Ireland after Brexit.
He said: "Prime Minister, go to Brussels and act in the interests of all citizens across the United Kingdom and negotiate to keep us in the single market and customs union, that will command the majority in the House of Commons".
The mammoth three-hour meeting came a day after eight Brexit-supporting ministers took the unusual step of meeting over pizzas in the office of Commons Leader Andrea Leadsom to discuss their concerns about the PM's stance.
LONDON - Theresa May is struggling to hold her government together as senior members of her cabinet openly meet to plot against her.
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The Prime Minister was urged to "go to Brussels and act in the interest of citizens across the EU, negotiate for us to keep us in the single market and customs union".
The prime minister heads to Brussels today with little expectation of unlocking the further progress on Brexit that had been hoped for this month.
International Development Secretary Penny Mordaunt told Sky News: "No-one is planning on resigning".
He conceded "we are not so far" from a deal but cautioned that "now we must accelerate the work".
"We are approaching the end of negotiations".
"That has got to be gripped, and I think the sooner it is gripped the better".
And Mrs Leadsom herself said: "The Prime Minister is doing a very, very complicated job and I'm fully supporting her in getting that done".
Ministers arriving for Cabinet on Tuesday morning were offered sausage and bacon sandwiches.
"Remain" MPs expressed anger after Brexit Secretary Dominic Raab published a new letter suggesting MPs would only be allowed to vote for or against May's final Brexit plan.
Although Barnier said the two sides need "much more time" to work "calmly and patiently" on a deal.
Germany's Europe minister, Michael Roth, said Berlin's message to Mrs May was: "Take responsibility and be constructive".
"The British side needs support".