Theresa May’s Brexit ordeal was thrown into sharp relief on Wednesday, as Conservative Eurosceptics plotted to oust the UK prime minister even as hopes rose in London and Brussels that an exit deal could be struck in November.

Mrs May’s allies privately admit her bigger challenge now is not securing a Brexit agreement in Brussels, but rather fending off an increasingly hostile Eurosceptic faction in her party that wants to stop her at all costs.

Hopes are rising in Downing Street that Mrs May can achieve both tasks, with aides eyeing a possible Brussels summit in mid-November for the signing of a Brexit deal, including key elements of her compromise Chequers plan.

Mrs May was delighted when Jean-Claude Juncker, European Commission president, echoed the Chequers phrase “free trade area” to describe the basis for a future UK-EU trade deal, rather than a simple “free trade agreement”.

Mr Juncker said in a speech to the European Parliament that the EU would not accept the break-up of its single market, but added: “I welcome Prime Minister May’s proposal to develop an ambitious new partnership for the future, after Brexit.”

In a largely positive speech, he added: “We agree with the statement made in Chequers that the starting point for such a partnership should be a free trade area between the UK and the EU.”

Mrs May approvingly quoted Mr Juncker’s words at the weekly prime minister’s question time in parliament and reflected the more positive tone of rhetoric between London and Brussels when she said to the EU: “You will never be an ordinary third party for us.”

The unusually warm exchanges reinforced the dawning realisation for many Conservative Eurosceptics that Mrs May may be about to strike a Brexit deal — but not on terms they can accept.

Difficult questions about Britain’s future trading relationship with the EU would be kicked into the second phase of talks — after Brexit in March 2019 — with both sides eyeing “aspirational” commitments to secure something better than a standard free trade deal.

“The only real outstanding issue is the Northern Ireland border and nobody seriously thinks that will stop a deal,” said one senior British official. “It will be difficult but in the end there will be an agreement.”

But the 60-70 members of the pro-Brexit European Research Group of Conservative MPs, led by Jacob Rees-Mogg, remain implacably opposed to Mrs May’s strategy and claim her Chequers plan for a “free trade area” would leave Britain as a “vassal state”, tied to Brussels’ rules on goods and agriculture.

However Mrs May’s critics have yet to articulate a comprehensive alternative plan, let alone work out how best to halt the prime minister if she insists on delivering what they believe will be “Brexit in name only”.

On Tuesday night around 50 ERG members met for private talks — later circulated to the media — detailing how around five of their number wanted to topple Mrs May.

Comments at the meeting were said to include “everyone I know says she has to go”, “she’s a disaster” and “this can’t go on”.

But by the time dawn broke on Wednesday the plot against Mrs May was already running into the sand, as it became clear the Tory hardliners did not yet have anything like the numbers to move against Mrs May.

The prime minister can so far count herself lucky in her enemies. Not only has the ERG so far failed to deliver on its promise of a rival Brexit white paper, the handful of leadership plotters were quickly denounced by fellow Eurosceptics.

“All I would say to them is: Stop it, it’s just stupid,” said Iain Duncan Smith, the Eurosceptic former Tory leader. “If you’ve got nothing better to do, go and find yourself some work, because that’s the best cure for stupid personality.”

Mr Rees-Mogg, launching an ERG paper proposing how a Canada-style trade deal was compatible with an open border in Ireland, said he wanted Mrs May to change tack on Brexit, but that she was a “fantastically dutiful prime minister”.

The plotters face two big problems. First they would have to assemble 48 Tory MPs to trigger a vote of no-confidence in the prime minister — a target that would theoretically be within reach in the event a disparate group of Eurosceptics decided to mobilise en masse.

The second problem is that Mrs May’s spokesman confirmed she would contest any vote of confidence, and her critics would then have to persuade half of the parliamentary party — or 158 MPs — to oust her.

One member of the ERG said: “I honestly doubt whether there are sufficient numbers to stage a coup. Every time there’s talk of a rebellion you have to halve the numbers and then halve them again in terms of the ones who will actually go through with it.”

The bigger test for Mrs May is whether she can sell her Brexit deal to the Commons in the “meaningful vote” that will determine the future state of Britain’s relationship with Europe — and her premiership.

Eurosceptic MPs say her Chequers plan is so despised by all sides of the Commons that it is certain to fail, and Downing Street acknowledges that pulling off a victory in such circumstances would be the biggest test of Mrs May’s career.

But Downing Street believes that ultimately her critics will step back from the cliff edge, knowing that if they vote down a Brexit deal it could unleash such political chaos that Brexit would not happen at all.