Rishi Sunak was “absolutely right” to issue a warning that immigration could “overwhelm” Europe, one of his ministers has claimed.

The prime minister made the remark during a right-wing Conservative event in Italy on Saturday, claiming “enemies” could use immigration as a “weapon” by “deliberately driving people to our shores to try to destabilise our society”.

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Questioned about the comment on Sky News’ Sunday Morning with Trevor Phillips, Deputy Prime Minister Oliver Dowden said: “I think the prime minister is absolutely right to issue this warning, and indeed it’s not just a warning, it’s something that we have seen elsewhere.

“We have seen the weaponisation of migration, for example, in the conduct of Belarus in relation to Poland, there’s been warnings from Finland in respect of the conduct of Russia.”

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Mr Dowden also claimed there was a “broader point” to the prime minister’s remarks, “which is that we do have to reassure people that we have got control of our borders and we cannot have this unsustainable situation where we’re enriching people smugglers – the worst people on the earth – through allowing this trade in human beings across the Atlantic, across the Channel”.

Pushed by Trevor Phillips over whether the language was “extreme” and could be hurtful to people living in the UK with an immigrant background, the deputy prime minister said he didn’t think Mr Sunak “meant it in that way at all”.

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He added: “What the prime minister meant, and indeed I think it is incumbent on mainstream politicians like myself and the prime minister and the Conservative Party, to make sure we deal with legitimate public concerns about uncontrolled migration.”

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The latest controversial discussion on immigration comes in a week where the government faced a battle within its own party to push forward with its Rwanda plan, which would see asylum seekers arriving in the UK via small boat crossings deported to the African nation.

Ministers managed to win a vote in the Commons in support of its Rwanda bill that seeks to allay the concerns of the Supreme Court over the scheme, following it ruling the policy unlawful in the courts last month.

While Mr Sunak secured the support of the more centrist wing of his party for the bill, some on the right criticised it for not going far enough in disapplying human rights laws, and 29 of them abstained on the vote.

The bill will now return to the Commons for its next parliamentary stage, and only 27 Tory MPs would need to vote against it for it to fall.

Mr Dowden did not rule out making changes to the legislation in order to get right-wing Tories onside – even though that would risk alienating the so-called One Nation centrists.

But he did claim the offer on the table was “the best thing we can get”.

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The deputy prime minister told Trevor Phillips: “We will listen to our colleagues about how we can improve this legislation, of course we will.

“[But] I think this is a good piece of legislation that does the job, which is about ensuring that we control migration.”

He added: “Of course we don’t rule out amendments and of course we will engage with that – that’s what happens with any piece of parliamentary legislation.

“What I would say, though, is that the prime minister is a pretty rigorous person, he’s looked through this very carefully, turned it upside down, shaken it around, he’s pretty sure this is the best thing we can get.

“But, of course, if there are other ways of improving it, we’ll be open to doing that.”

Labour’s shadow health secretary, Wes Streeting, attacked both the Rwanda plan and the prime minister’s remarks over the weekend.

He told Trevor Phillips the speech was “a stunning admission of failure”, adding: “This is the guy who says, ‘one of my priorities is to stop the boats [and] I’m doing a really great job’, then the next minute he’s off to Italy to say the quiet bit out loud to all his right wing chums across Europe – which is he’s failed.

“He’s failed both to manage an immigration system that works for legal migration, he’s failed to tackle illegal migration and meanwhile he is pulling his own party apart with this spectacular circus we saw this week over a policy that will not work and is costing us hundreds of millions of pounds already”

Mr Streeting said the government has “got to get a grip on this”, and claimed his party’s policy – including closer working with France and additional staff to tackle backlogs in the Home Office – was not “gimmicky” but “practical” for reducing illegal migration and “better managing” legal migration.

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