In calling for Theresa May to abandon her ambition to reduce immigration to the “tens of thousands” a year, the Confederation of British Industry (CBI) is reflecting public opinion better than the prime minister. A YouGov survey of an exceptionally large sample of 10,000 electors finds that 70% are happy for EU citizens with a job or university place in the UK to be free to come to Britain. Only one voter in six wants a sharp reduction in the numbers of workers and students coming from the EU.

The survey, which was commissioned by the People’s Vote campaign, sought to explore underlying public attitudes to Brexit, as negotiations between London and Brussels head towards their climax. One of the trickiest issues concerns freedom of movement. To many in the EU, this sits alongside the freedom to trade goods, services and capital across national borders: the four freedoms are indivisible. Plainly, Britain cannot expect the right to drastically curb immigration from the EU while seeking the right to keep trade flowing freely and without friction.

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However, freedom of movement has never been absolute. A number of countries recognise the right to free movement for labour, but not the complete right to free movement of people. They welcome those from the rest of the EU who come to work or study, but not to stay if they don’t have a job or college place.

YouGov has found only limited support for completely unfettered free movement – but clear majority support for the kind of qualified freedom advocated by the CBI.

Even leave voters tend to support qualified freedom of movement; while only 4% favour complete freedom, 58% think workers and students should be free to come to Britain. Just 31% of leave voters want a sharp reduction in EU immigration.

What lies behind this public rejection of the kind of draconian target that May has embraced since she was home secretary? Two further YouGov findings help to explain what is going on.

First, when voters are asked which is more important for Britain, the right to control immigration from the EU or the ability to trade freely with the EU without tariffs or other restrictions, a majority of almost two to one (50% to 29%) plump for trade. Leave voters veer in the opposite direction, but not as decisively as might have been expected: 52% reckon control of immigration is more important, but a significant minority, 30%, would give priority to the economic imperative of frictionless trade.

Second, with recent stories of the NHS finding it hard to recruit doctors and nurses from abroad, voters are in no mood to shut the door. Seventy-three per cent (including 64% of leave voters) agree that “we need nurses and doctors from overseas to keep the NHS running”. Only 16% of the public think “it will be easy to replace the overseas doctors that currently work in our NHS with UK nationals”.

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There is a third factor. In recent years, YouGov and Ipsos Mori have found that many voters, and especially those most critical of the EU, believe that far more EU immigrants are receiving out-of-work welfare benefits than those reported in government statistics. A big part of the anti-immigration mood flowed from a belief that hundreds of thousands of EU nationals, especially from eastern Europe, regarded the UK as a soft touch and came here as “welfare tourists”.

Spasmodic attempts to show that this was not true came to nothing. But it is increasingly clear that the great majority of British voters are happy to admit economically useful EU immigrants, so long as this freedom is not extended to those who do not have, or do not quickly obtain, a job or university place.

Michel Barnier has been reported this week as being willing to soften his stance on freedom of movement. The CBI has suggested one way forward. The British public are up for a qualified freedom of movement. Can May’s “tens of thousands” target last much longer?

• Peter Kellner is a former president of YouGov