Nearly half of Britons are doubtful about the Government's chances of meeting its controversial immigration target even after leaving the EU, according to a report.
There is scepticism concerning the prospects of bringing net migration below 100,000, a survey for British Future suggested.
Sunder Katwala, director of the think-tank, said: "Public trust in governments' competence to manage immigration - including meeting its own targets - is at rock bottom."
Meanwhile, the Institute for Public Policy Research claimed in a separate report that the NHS "would collapse" without workers from the European Union, and called for them to be offered automatic citizenship as part of efforts to head off a "brain drain".
The first official immigration figures since the Brexit referendum will be published on Thursday.
In the previous set of figures, net migration - the difference between the numbers arriving to and leaving the UK - was an estimated 333,000 in the year to December.
Britain is expected to seek to introduce controls on free movement rules following the referendum outcome, but details of the system have yet to be outlined.
A poll of more than 2,000 adults for British Future examined attitudes regarding the aim to reduce net migration to five figures.
It found that 37% think the UK is likely to meet the target in the next five years, compared to 44% who consider the chances of reaching the aim unlikely.
The report also said:
::12% of people would like to see a reduction in the numbers of highly skilled workers migrating to Britain, while nearly four times as many (46%) would like to see more arriving.
::People are "less positive" about low-skilled workers moving to the UK, with nearly four in 10 (38%) happy for numbers to stay the same or increase but six in 10 (62%) preferring a reduction.
Mr Katwala said: "The Brexit shake-up could be an opportunity to get immigration policy right - to restore trust in a system that works, and public consent for the immigration that we have."
Elsewhere, an IPPR paper cited figures suggesting there are currently around 55,000 EU nationals working in the English NHS, and one in 10 of the UK's registered doctors is an EU national.
"Without them, the NHS would collapse," the report said, recommending that the Government "makes a particularly generous citizenship" offer to NHS workers.
It said all EU nationals who work for, or as locums in, the NHS should be eligible to apply for British citizenship.
The study called for a waiver of fees, English language and "life in the UK" tests, and residency requirements for all staff employed in public healthcare.
IPPR research fellow Chris Murray said: "Britain can ill afford to lose the talents of many EU migrants who have made Britain their home.
"EU migrants who are already here should get indefinite leave to remain. In particular, EU NHS workers should get the automatic right to citizenship. If they left, it would be a crisis for the NHS."
The status of more than three million EU migrants already in Britain came under scrutiny after the Brexit vote.
The Government has said it "fully expects" that the legal status of EU migrants living in this country will be "properly protected" when the UK leaves.
A Government spokesman said: "The Government is committed to reducing net migration to sustainable levels.
"We have been clear that a sustainable level means tens of thousands, not hundreds of thousands."