Theresa May will move to seize back the initiative from mutinous Tory MPs on Monday by promoting her Brexit deal with a defiant speech to business leaders, even as critics in Westminster scramble to trigger a no-confidence vote in her leadership.

As she enters perhaps the most perilous week of her premiership, May will insist at the CBI annual conference in London that her deal delivers on the central demand of voters in the 2016 referendum, by allowing the UK to control immigration.

“Getting back full control of our borders is an issue of great importance to the British people,” she will say, adding that EU citizens will no longer be able to “jump the queue ahead of engineers from Sydney or software developers from Delhi”.

Downing Street hopes that by stressing what it regards as the positive aspects of the deal reached with Brussels last week, May can win over doubters in her party and undercut Tory rightwingers calling for her to be ousted.

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She will also travel to Brussels this week to hammer out the final details of the political declaration on Britain’s future relationship in person with Jean-Claude Juncker, the president of the European commission – in the hope the whole package, which also includes the withdrawal agreement containing the backstop, can be approved at a special EU summit next Sunday.

But as May presses ahead with Downing Street’s carefully choreographed plan to seal – and sell – the agreement, she continues to come under relentless fire from her own party.

MPs from the Brexit-backing European Research Group insisted on Sunday they were confident of corralling the 48 MPs they need to write letters to force a vote of no confidence in the prime minister.

“I would expect so,” said the ERG’s Crispin Blunt, asked whether the target would be hit. He said: “It’s in the prime minister’s interests that it’s got out of the way.”

Speaking on Sophy Ridge on Sunday on Sky News, May said internal critics thinking of replacing her as Conservative leader should think again. The prime minister said: “It is not going to make the [Brexit] negotiations any easier and it won’t change the parliamentary arithmetic.”

Asked by Ridge whether she had considered stepping down, May said: “No, I haven’t. Of course it has been a tough week. Actually these negotiations have been tough right from the start, but they were always going to get even more difficult right toward the end when we are coming to that conclusion.”

Play Video 1:16 Threshold for confidence vote not reached, says prime minister – video

May’s allies have suggested even if she scraped through with a narrow margin, she would remain in Downing Street to complete the task of taking Britain out of the EU. But Blunt said: “She would have a decision to make about whether that gives her enough authority to carry on.”

ERG chair Jacob Rees-Mogg announced at a press conference in Westminster on Thursday that he was submitting his own no-confidence letter, sparking suggestions the total could be reached within hours, but Friday passed without any announcement from the chair of the backbench 1922 Committee, Sir Graham Brady.

Brady told BBC 5 Live on Sunday he would inform the prime minister first, if the total was reached – and he would then expect a vote to take place swiftly. “It ought to be a test of opinion very quickly, in order to clear the air and get it resolved,” he said.

He said the feverish speculation about the number of letters he had received had spread beyond the corridors of the Palace of Westminster. “I get asked it in the supermarket, I get asked it in the street,” he said. “I’ve become very nervous about counting, or saying numbers, in case people think I’m saying something that I’m not.”

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Senior Brexiters advanced a number of explanations for a possible shortfall on Sunday, including the idea that some MPs were waiting until Monday to deliver their letters in person, because they feared an email would be hacked.

As the ERG exhorts its members to submit their letters, potential leadership challengers continue to jostle for position.

Friends of Boris Johnson denied reports that he had held secret meetings with the former Brexit secretary David Davis in a bid to strike a deal about who would run. They insisted Johnson had not discussed handing the job of chancellor to Rees-Mogg. Neither Johnson nor Davis has joined public calls for May to be ousted.

Davis’s successor, and former chief of staff, Dominic Raab, is also being widely touted as a leadership candidate after resigning last week in protest at the deal, as is Amber Rudd, who returned to cabinet on Friday as work and pensions secretary.

Those Brexiters who have remained in May’s cabinet appeared to be backing away from the idea of presenting her with an ultimatum of reopening talks on the Irish backstop or facing fresh resignations.

Penny Mordaunt, the development secretary, is now not intending to attend a meeting being convened by Andrea Leadsom to compare notes on how the Brexit deal could be improved.

Friends of Michael Gove said that while he was keen to consult cabinet colleagues, May had already rejected his pleas to seek to rewrite the backstop last Thursday when he turned down the job of Brexit secretary. “It doesn’t seem likely she’s going to budge,” said a source.

Downing Street sources confirmed that they regarded the withdrawal agreement as complete and all their focus in the final few days of talks would be on fleshing out the seven-page political declaration.

The Democratic Unionist party, whose MPs prop up May’s minority government, said the UK had to push for a better deal that did not put “a trade border down the Irish sea”. Nigel Dodds, the party’s leader in Westminster, said: “Even Jeremy Corbyn gets it, although nationalists and republicans here are desperate for him to stop saying it.”

May hopes to receive a friendly hearing at the CBI’s annual conference, which will also be addressed by the Labour leader, with business leaders keen to avoid the risk of no deal.

The CBI’s president, John Allan, will urge politicians contemplating voting against the deal to “listen to the businesses in your constituencies – and everyone who depends upon them”.

“While companies in this room would be the first to say that it is not perfect, it does open a route to a long-term trade arrangement and unlocks transition – the very least that companies need to prepare for Brexit.

“And most importantly, it avoids the wrecking ball that would be a no-deal departure,” he will say, claiming that 80% of firms have already cut or postponed investment because of the risk of a no-deal exit.

The prime minister will tell her audience in Greenwich: “I have always had a very clear sense of the outcomes I wanted to deliver for people in these negotiations. Control over our borders, by bringing an end to free movement, once and for all. Control of our money, so we can decide for ourselves how to spend it, and can do so on priorities like our NHS.

“Control of our laws, by ending the jurisdiction of the European court of justice in the United Kingdom and ensuring that our laws are made and enforced here in this country.”