There has been growing speculation at Westminster that Mr Jenrick is keen to position himself as the flagbearer of the right in the likely event of a general election defeat.
He was asked on Sunday if he was talks with the New Conservatives group – 2017 and 2019 MPs who have led the rebellion over the bill – about lining himself up to be a future leader.
“What I am trying to do is to make an argument and part of that is around illegal migration. I think that for too long, too few politicians have gone and argued that we need to take the most robust action,” he told GB News.
Pressed again if he would stand in the future as leader, Mr Jenrick said: “Well, look, I’m not ruling it out. But that’s not my intention here. What I really want to do is make and hopefully win this central argument for the Conservative party’s future.”
Mr Jenrick attacked the government following a report claiming that 16,000 asylum seekers – including some who had arrived in the UK on small boats – have been allowed to work here.
“I think that’s wrong … it just creates a pull factor to the UK,” said Mr Jenrick, who has adopted increasingly harsh anti-immigration rhetoric since quitting in early December.
“Almost everyone who comes here is either an economic migrant or a sort of asylum shopper, because they’re coming from safe countries like France and choosing to come to the UK because they think life is better here or a softer touch.”
A close ally of ex-home secretary Suella Braverman – herself thought to be keen on the leadership – has told people that business secretary Kemi Badenoch should be installed as leader if there is a push against Mr Sunak, according to the Sunday Times.
The Conservatives’ election strategy Isaac Levido has appealed for unity as the party continues to squabble over the totemic Rwanda policy.
While only 11 MPs voted against the Rwanda bill during a crunch vote earlier this week, including Mr Jenrick, some rebels have warned they could try to change the bill again when it comes back from the Lords.
It comes as senior Tory peer Nicky Morgan warned Mr Sunak to curb his “will of the people” rhetoric, as the government shapes up for a major battle with the House of Lords.
The prime minister has urged peers to “crack on” with approving the Rwanda bill and not frustrate the “will of the people” – saying he wanted to get the deportation flights “up and running” as soon as possible.
“I think the [Rwanda] bill will go through the House of Lords”
Conservative peer Nicky Morgan says she thinks the government will get its legislation passed but that Rishi Sunak using “will of the people language” is not a “happy precedent”
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Baroness Morgan warned Mr Sunak not to put the Lords under “too much pressure” as he pushes to get it through parliament in time for a push to get flights under way in spring.
“I would just say to No 10 the last time a prime minister who used the ‘will of the people’ language, it wasn’t a happy precedent,” she told BBC’s Sunday with Laura Kuenssberg, comparing him to Boris Johnson Brexit battles.
“The thing to do is give it time,” she said. “It’s a scrutinising chamber, there’s lots of people – lawyers, diplomats other, who will say, ‘Are you sure about this?’ … That is what a scrutinising chamber is about.”
The influential peer Lord Goldsmith has laid a motion in the Lords which is aimed at halting the ratification of the Rwanda treaty. “It should not be ratified – it should be delayed,” he said.
The motion by the Labour peer, who was Tony Blair’s attorney general during the Iraq conflict, is set to be debated on Monday. It demands that the government demonstrates Rwanda is safe for asylum seekers.
The Liberal Democrats are expected to table a “fatal motion” aimed at killing the bill, but it is unlikely to succeed.