A Home Office minister has refused to say if immigration will be lower after Brexit and admitted the Conservatives do not yet have a policy.

In an excruciating exchange, Victoria Atkins failed – four times – to answer the question, despite cutting the numbers of migrants being a central pledge of the Vote Leave referendum campaign.

Asked if the Tories wanted immigration to be higher or lower than it is now, she said: “We want to have immigration that suits the needs of the country.”

Pressed again, Ms Atkins – who stood in for Priti Patel, the home secretary, who declined the interview – said only that immigration was “one of the key issues that’s raised on the doorstep”.

She then said: “We want to have a points-based system that focuses on bringing the brightest and the best into the UK, precisely so that we can attract doctors, nurses and so on into the NHS.”

But Nick Robinson, the Radio 4 interviewer, pointed out that was “a slogan, it means nothing on its own”, adding: “It’s like saying a food-based diet.

“It just means there will be points on a system. It doesn’t tell me who will come, how many will come.”

The minister admitted her party was only about to “start having these debates about what we want our immigration policy to look like”.

The refusal to answer the question will blunt Conservative attacks on Labour over immigration, with Jeremy Corbyn under pressure to say if he backs continued free movement of EU citizens after Brexit.

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The policy was backed overwhelmingly by delegates at Labour’s conference in September – overturning the party’s stance in 2017 – but it is unclear if it will be in this year’s manifesto.

In the summer, Boris Johnson ripped up Theresa May’s tough immigration plans – including her infamous pledge to slash annual numbers to “tens of thousands” – and ordered officials to explore more liberal rules.

Advisers will work up plans for “an Australian-style points-based system”, but no details have been put forward.

Already effectively used for migrants coming from outside the EU, many believe it could increase the numbers coming to the UK if based on education levels or needs for skills.

The existing immigration bill, now paused by the election, is meant to prepare the UK for life after Brexit and an end to free movement of EU citizens.

It was attacked for leaving businesses, the NHS and other employers in the dark about a mooted £30,000 salary threshold – a significant clampdown – after cabinet infighting.

In the interview, Ms Atkins was also asked if she wished to apologise for the Tories missing their immigration target for 37 months in a row.