Theresa May has warned her Tory opponents their alternate Brexit plans would not make their problems disappear as she faced a continuing gruelling battle to win over sceptical MPs.

After bolstering her Cabinet by bringing back former home secretary Amber Rudd, Mrs May warned that the so-called Norway and Canada models favoured by her most hardline critics would not solve the issue of the Irish border backstop.

It came after a tumultuous week that saw the departure of ministers including Brexit secretary Dominic Raab and work and pensions secretary Esther McVey, and the launch of a high-profile insurrection on the back benches to remove her from office.

Ms Rudd pleaded with Tory rebels to back the Prime Minister and her Brexit deal on her return to Government as the new Work and Pensions Secretary, saying: “This is not a time for changing our leader.”

But the Prime Minister’s critics believe they have the numbers required to trigger a confidence vote within days.

And Brexit-supporting ministers led by Andrea Leadsom are reportedly set to work together on measures to make the deal more acceptable to them.

But in an interview with the Daily Mail, Mrs May bluntly told them their alternatives would not work.

She said: “People say ‘If you could only just do something slightly different, have a Norway model or a Canada model, this backstop issue would go away’. It would not. That issue is still going to be there.

“Some politicians get so embroiled in the intricacies of their argument they forget it is not about this theory or that theory, or does it make me look good.’

New Brexit Secretary Stephen Barclay (Chris McAndrew/UK Parliament/(Attribution 3.0 Unported (CC BY 3.0)/PA)
New Brexit Secretary Stephen Barclay (Chris McAndrew/UK Parliament/(Attribution 3.0 Unported (CC BY 3.0)/PA)

Mrs May appointed Stephen Barclay as her new Brexit Secretary, the third this year, but the role has been further downgraded as the Prime Minister will in future take sole control of negotiations on EU withdrawal.

Leave-supporting Mr Barclay’s job will be limited to the domestic delivery of EU withdrawal, preparations for Brexit either with or without a deal and shepherding legislation through Parliament.

He said: “We now need to keep up the momentum to finalise the withdrawal agreement and outline political declaration and deliver a Brexit that works for the whole UK.”

The reshuffle came just hours after Michael Gove offered the Prime Minister a lifeline by staying on in his Cabinet role.

Mr Barclay’s appointment came after Mr Gove reportedly turned down the post, saying he would only take it if he could renegotiate the EU withdrawal agreement.

Downing Street declined to say whether the Brexit Secretary post had been offered to anyone else before the North-East Cambridgeshire MP, saying only: “He was the Prime Minister’s choice for the job.”

Party chairman Brandon Lewis said the Prime Minister is “prepared for anything”, when asked on BBC Radio 4’s Today programme if she was prepared for a leadership challenge.

He added: “I think that whatever deal, when we see various different organisations and groups talk about different ways of leaving the EU – Norway, Canada, all of these things – what we have to be clear about … is all of those require a backstop.”

Ireland’s deputy premier has warned British politicians to sign up to the current Brexit agreement or risk crashing out of the EU without a deal.

Simon Coveney said people were “too quick” to write off Mrs May, saying she has shown “resilience, courage” and a “reality check” to Brexiteers in Westminster.

Speaking ahead of Fine Gael’s Ard Fheis (conference), he warned it “will be difficult” to find an alternative deal that will prevent a hard border.

“If we are forced into that situation we will have to look at ways in which that can be avoided but that will not be easy,” he said.

“It is a far more sensible approach to support what is being negotiated over the last two years as a way that we know provides the guarantees that we have looked for and insisted on that there will be no physical border or infrastructure or checks or controls and that is an issue that goes beyond commerce and trade.”

David Davis, who was Brexit secretary before Mr Raab, had earlier used a series of tweets to again urge Mrs May to seek a looser deal with the EU.

He said the United States would be ready to start negotiating a free trade deal with Britain “immediately” after Brexit if Theresa May’s plan is ditched, while making a trip across the Atlantic.

The former Brexit secretary revealed he had spent several days in Washington talking to “US Government Trade and Treasury officials” while Westminster had been reeling from high-profile resignations and questions over the Prime Minister’s future.

“They have already started on the procedures to allow negotiations to start immediately once we leave the EU in March,” he said.

“This will not be possible if we accept the Government’s proposed deal with Europe, which will block every avenue of negotiation with America.

“We have to have a Canada+++ deal to allow us to have a free trade deal with America.”