Ireland is pledging emergency legislation enabling it to send asylum seekers back to the UK.
More than 80% of recent arrivals in the republic came via the land border with Northern Ireland, Irish justice minister Helen McEntee told a parliamentary committee last week.
Rishi Sunak told Sky News it showed the UK’s Rwanda scheme was already working as a deterrent after it finally became law last week.
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Ireland’s deputy prime minister has said the threat of deportation to Rwanda is causing migrants to head for Ireland instead of the UK.
Micheal Martin said the policy was already affecting Ireland because people are “fearful” of staying in the UK.
The former taoiseach told The Daily Telegraph: “Maybe that’s the impact it was designed to have.”
Simon Harris, Ireland’s latest leader, has asked Ms McEntee to “bring proposals to cabinet to amend existing law regarding the designation of safe ‘third countries’ and allowing the return of inadmissible international protection applicants to the UK”, a spokesman said.
Ms McEntee said she will be meeting UK Home Secretary James Cleverly in London on Monday.
“There are many reasons why we have seen an increase in migration towards Ireland,” she told RTE.
“My focus as minister for justice is making sure that we have an effective immigration structure and system.
“That’s why I’m introducing fast processing, that’s why I’ll have emergency legislation at cabinet this week to make sure that we can effectively return people to the UK, and that’s why I’ll be meeting with the home secretary to raise these issues on Monday.”
People are now “worried” about coming to the UK, Rishi Sunak has said.
He told Sky News: “If people come to our country illegally, but know that they won’t be able to stay here, they are much less likely to come, and that’s why the Rwanda scheme is so important.”
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Mr Sunak said the comments from Irish politicians show that “illegal migration is a global challenge”.
“[That] is why you’re seeing multiple countries talk about doing third country partnerships, looking at novel ways to solve this problem, and I believe [they] will follow where the UK has led,” he said.
Shadow minister Wes Streeting said it was unlikely a Labour government would bring people back from Rwanda if some are sent there.
“Once people are settled in Rwanda, they’re settled in Rwanda,” he told Sky News, adding it was doubtful that Labour would “unpick that situation”.
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Regarding illegal migration in general, he said it required “putting the money that’s gone to Rwanda into the National Crime Agency so we can have proper cross-border policing to tackle the criminal gangs, speeding up the processing of decision-making, making sure we’ve got serious returns agreements with other countries”.
He added: “Those are solutions that can work.”
