More than eight out of 10 people in the UK believe EU migrants already living in Britain should be allowed to remain after Brexit, including 77% of Leave voters.

The figures are revealed in new poll for the British Future thinktank which wants a “national conversation” on immigration as part of a comprehensive review of a system in which, it says, “the public has lost all confidence”.

In its new report, “What next after Brexit? Immigration and integration in post-referendum Britain”, British Future claims that its ICM poll confirms that the majority of people in post-referendum Britain fall into what calls the “anxious middle” – while concerned about the pressures of high migration, they also accept the benefits that migrants bring to the economy and wider society.

Before the release of the latest immigration statistics on Thursday, the first since June’s referendum, the poll suggests that the public would be happy to see some flows of immigration increase but want reductions in other areas, notably the number of unskilled workers.

It finds that three-quarters of those polled agree with the call for a “sensible policy to manage immigration that controls who comes to the UK, but still keeps the immigration that is good for our economy and society, and maintains Britain’s tradition of offering sanctuary to refugees who need protection”.

Other key findings include:

■ 84% say EU citizens already living in the UK should be able to stay. This includes a majority of both Leave voters (77%) and Ukip supporters (78%);

■ Only 12% want to cut the number of highly skilled workers migrating to Britain; nearly half (46%) would like to see an increase, with 42% saying that it should stay the same;

■ Almost two-thirds (62%) want numbers of low-skilled workers reduced.

British Future argues that opening up a public debate about immigration now would bring about a new consensus on the divisive issue.

Brexit presents a chance to rebuild public trust in the immigration system Jill Rutter, British Future

“There are sure to be changes to immigration policy once we know what shape Brexit takes,” said Jill Rutter, director of strategy for British Future. “That will bring challenges but it also presents an opportunity – for a comprehensive review of a system that is widely believed to be failing and in which the public has lost all confidence. Rebuilding public trust, in an immigration system that is competent, effective and fair, must be part of this process. Engaging the public in the decisions we make, through a national conversation on immigration, would help to start rebuilding that trust.

“It will also cut through an overheated, polarised debate to reveal the moderate core of public opinion on immigration. Most people have more nuanced views than those found in our public discourse. Given the choice, voters would be content with much immigration staying the same and some of it increasing, if they had faith in the system and could see reductions in other areas.”

The poll reveals that the public is split roughly down the middle on refugees. Just over half (53%) think the number of refugees offered protection should be reduced while 33% think the country should offer sanctuary to about the same number of refugees as it does currently and 14% would like it to take more.

The poll also suggests the public makes clear distinctions between different immigrant worker groups.

Only a quarter of people want fewer migrant care-workers, with 27% saying they would like more and 48% saying the number should stay the same.

Four in 10 welcome more migrant engineers, compared to only 17% who want fewer. More people said they would like to see more migrant IT professionals, doctors, nurses and scientists than would prefer a cut in numbers.

It also confirms that even Leave voters are keen on some forms of migration.Only 15% of Leave voters want a reduction in the numbers of highly skilled workers migrating to Britain, while 45% want an increase and 40% want numbers to remain the same.

When asked about migrant IT specialists, engineers, scientists, care workers, doctors and nurses, the majority of Leave voters wanted an increase or the numbers to remain the same. Only when asked about unskilled workers, construction workers and hospitality staff did they prefer to cut numbers.

Thursday’s immigration statistics are expected to show a rise in EU nationals applying for British citizenship. Experts suggest this may reflect worries about their status in a post-Brexit UK.