Conservative Brexiteers are giving Theresa May an ultimatum, that if her negotiating strategy fails she must accept plans for a Canada-style trade deal or face a leadership challenge.

Senior figures say it would be impossible for her to try to further negotiate on her Chequers proposals if they are rejected by parliament or the EU.

Instead, she would be told only full acceptance of the kind of arrangement put forward by Boris Johnson, David Davis, Jacob Rees-Mogg and others would allow her to avoid a Tory vote of no confidence.

The prime minister will meet her full cabinet on Tuesday and is expected to present further tweaks to her proposals relating to future customs arrangements and EU regulation in a bid to secure a deal with Brussels.

There is also increasing speculation that Ms May could try to stay on in her job, either to renegotiate if initial talks fail to bear fruit, or to fully deliver Brexit if a deal is reached.

It follows a Conservative conference that ended on a positive note as the prime minister sought to refocus her party on domestic policy, but the agenda was always going to shift back to Brexit with another critical European summit days away.

A senior Conservative figure told The Independent: “The prime minister has three potential courses of action if her deal is voted down by parliament or rejected in Brussels.

“In that case she may be tempted to try to return to the negotiating table and develop Chequers with further concessions, or alternatively go down the route of joining the EEA. Neither of those are going to be acceptable to the party if Chequers has already been rejected.

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“If she does try to go there, colleagues are going to push back hard and given what would represent the collapse of her strategy, it will be her whole leadership in question.”

The individual explained that Ms May would only be allowed to avoid some kind of challenge if she agrees to pursuing the kind of Canada-style free trade deal that European Council president Donald Tusk said is on offer this week.

If she does try to go there, colleagues are going to push back hard and given what would represent the collapse of her strategy, it will be her whole leadership in question Senior Conservative figure

The Conservative MP said: “If she undertakes to secure that deal, it would be the only credible way she could possibly stay in the job at that point.”

Ahead of Ms May’s conference speech, Conservative MP James Duddridge revealed he had sent a letter to the chairman of the Conservative 1922 backbench committee, Sir Graham Brady, calling for a leadership contest.

Under Conservative Party rules, the 1922 chair must call a no-confidence vote of the parliamentary party in the prime minister if he receives 48 letters from MPs. It is not known how many have been submitted up to now.

A Conservative backbencher confirmed the view that Ms May would have to pursue a Canada-style free trade deal or face a challenge if Chequers falls.

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(Photo by Oli SCARFF / AFP)OLI SCARFF/AFP/Getty Images OLI SCARFF AFP/Getty Theresa May dancing on to stage before major conference speech Britain's Prime Minister Theresa May dances a few steps as she takes the stage to give her keynote address on the fourth and final day of the Conservative Party Conference 2018 at the International Convention Centre in Birmingham, central England, on October 3, 2018. (Photo by Paul ELLIS / AFP)PAUL ELLIS/AFP/Getty Images PAUL ELLIS AFP/Getty Theresa May dancing on to stage before major conference speech BIRMINGHAM, ENGLAND - OCTOBER 03: British Prime Minister Theresa May dances as she walks out onto the stage to deliver her leader's speech during the final day of the Conservative Party Conference at The International Convention Centre on October 3, 2018 in Birmingham, England. Theresa May delivered her leader's speech to the 2018 Conservative Party Conference today. Appealing to the "decent, moderate and patriotic", she stated that the Conservative Party is for everyone who is willing to "work hard and do their best". This year's conference took place six months before the UK is due to leave the European Union, with divisions on how Brexit should be implemented apparent throughout. (Photo by Christopher Furlong/Getty Images) Christopher Furlong Getty Theresa May dancing on to stage before major conference speech Britain's Prime Minister Theresa May dances a few steps as she takes the stage to give her keynote address on the fourth and final day of the Conservative Party Conference 2018 at the International Convention Centre in Birmingham, central England, on October 3, 2018. (Photo by Oli SCARFF / AFP)OLI SCARFF/AFP/Getty Images OLI SCARFF AFP/Getty Theresa May dancing on to stage before major conference speech Britain's Prime Minister Theresa May dances a few steps as she takes the stage to give her keynote address on the fourth and final day of the Conservative Party Conference 2018 at the International Convention Centre in Birmingham, central England, on October 3, 2018. (Photo by Oli SCARFF / AFP)OLI SCARFF/AFP/Getty Images OLI SCARFF AFP/Getty Theresa May dancing on to stage before major conference speech Prime Minister Theresa May dances as she arrives on stage to make her speech at the Conservative Party annual conference at the International Convention Centre, Birmingham.PRESS ASSOCIATION Photo. Picture date: Wednesday October 3, 2018. See PA story TORY Main. Photo credit should read: Victoria Jones/PA Wire Victoria Jones PA Theresa May dancing on to stage before major conference speech Britain's Prime Minister Theresa May dances a few steps as she takes the stage to give her keynote address on the fourth and final day of the Conservative Party Conference 2018 at the International Convention Centre in Birmingham, central England, on October 3, 2018. (Photo by Oli SCARFF / AFP)OLI SCARFF/AFP/Getty Images OLI SCARFF AFP/Getty Theresa May dancing on to stage before major conference speech Britain's Prime Minister Theresa May dances a few steps as she takes the stage to give her keynote address on the fourth and final day of the Conservative Party Conference 2018 at the International Convention Centre in Birmingham, central England, on October 3, 2018. (Photo by Oli SCARFF / AFP)OLI SCARFF/AFP/Getty Images OLI SCARFF AFP/Getty Theresa May dancing on to stage before major conference speech Prime Minister Theresa May dances as she arrives on stage to make her speech at the Conservative Party annual conference at the International Convention Centre, Birmingham.PRESS ASSOCIATION Photo. Picture date: Wednesday October 3, 2018. See PA story TORY Main. Photo credit should read: Victoria Jones/PA Wire Victoria Jones PA Theresa May dancing on to stage before major conference speech Britain's Prime Minister Theresa May dances a few steps as she takes the stage to give her keynote address on the fourth and final day of the Conservative Party Conference 2018 at the International Convention Centre in Birmingham, central England, on October 3, 2018. (Photo by Paul ELLIS / AFP)PAUL ELLIS/AFP/Getty Images PAUL ELLIS AFP/Getty Theresa May dancing on to stage before major conference speech BIRMINGHAM, ENGLAND - OCTOBER 03: British Prime Minister Theresa May delivers her leader's speech during the final day of the Conservative Party Conference at The International Convention Centre on October 3, 2018 in Birmingham, England. Theresa May delivered her leader's speech to the 2018 Conservative Party Conference today. Appealing to the "decent, moderate and patriotic", she stated that the Conservative Party is for everyone who is willing to "work hard and do their best". This year's conference took place six months before the UK is due to leave the European Union, with divisions on how Brexit should be implemented apparent throughout. (Photo by Christopher Furlong/Getty Images) Christopher Furlong Getty

He said: “Of course she would. If her entire negotiating strategy falls apart how can she possibly keep trying to negotiate on it.

“Delivering a free trade deal as the only option left that could get through parliament after the death of Chequers, is the only thing she might stay on to do. Even then it would be in the face of calls to quit.”

A member of the cabinet told The Independent that it would not be unexpected for sections of the party to clash with the prime minister if Chequers fails.

“If you’re pursuing a strategy and the strategy is a failure, then people are going to ask the individual who advocated it and led it to take responsibility for the outcome,” the frontbencher said.

“It’s also true that if Chequers or whatever it ends up being called doesn’t work, then it’s hard to see what else other than a free trade deal will get through, either in Brussels or London.”

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The Independent reported last week how Ms May is preparing to tweak her approach to make it more saleable to the EU, accepting a customs arrangement more closely aligned to Brussels than previously thought and accepting more European regulation in future.

Parliament’s return on Tuesday gives her the first opportunity to gauge cabinet support, with several members known to be more inclined towards a Canada style-free trade deal.

Some ministers said ahead of conference that they were even pushing for the word “Chequers” to be removed from the government’s discourse on Brexit, with Ms May then not saying it in her keynote speech.

Johnson devoted his conference appearance to attacking Ms May’s plans, branding them a “cheat” and saying there is still time to adopt a Canada-style arrangement.

Advocates say it is the only way the UK can gain enough freedom from EU regulation and customs to sign effective free trade deals, but Downing Street argues it would mean splitting the UK, because Brussels has suggested it would necessitate Northern Ireland staying inside an EU customs union to keep open the border with the Republic – an argument Brexiteers reject.

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On Thursday, Mr Tusk said a Canada-style deal had always been on offer “from the very beginning”, increasing pressure on Ms May to change tack ahead of the summit in mid-October.

Looking beyond Brexit, Ms May has said she is in post for the “long term”, though since the botched 2017 election there has been an understanding inside and outside Downing Street that after withdrawal her leadership would come into question.

More recently, however, some MPs say they have detected signs that she thinks she really can stay on after Brexit.

One senior backbencher told The Independent: “There’s a real fear she is beginning to think she should stay, that she doesn’t want her time to be defined by Brexit.