Reports from employers about possible illegal workers are sometimes being ignored by the Immigration Service, a BBC investigation has found.
Immigration Service staff have not dealt with reports, agencies say
Recruitment agencies say that when they pass on concerns about fake papers, the response is often that nothing can be done because of a lack of resources.
"Failed asylum-seekers are the only priority," said John Tincey of the Immigration Service Union told the BBC.
The Home Office says it does tackle the issue but must prioritise resources.
Last month, the home secretary asked the public to inform on employers who take on illegal workers.
However, employment agencies told the BBC they felt the Immigration Service was not playing its part.
One agency, which recruits for the care and nursing sector, said that when three people were found to have false documents last year, the Immigration Service asked that they be kept in work so their whereabouts were known.
Five months later the service said it would not be taking action, because of a lack of resources.
Nursing and care agencies feel particularly concerned about this, as they are looking after vulnerable and elderly people.
An online survey of 425 recruitment agencies conducted by the recruitment industry body the Recruitment and Employment Confederation suggested three-quarters of agencies had spotted suspect documents.
But nearly half said the Immigration Service did not take decisive action.
"The reason we don't arrest these illegal immigrants is we're told not to by our managers," Mr Tincey told BBC Radio 4's Today programme.
"This has been the situation for a number of years because the priorities set by the Home Office are to arrest failed asylum-seekers, and other offences such as illegal working are very much on the back-burner."
Some workers felt their jobs "were on the line" unless they complied and "senior managers have criticised staff for actually spending time looking for illegal workers", he added.
"There is nobody to do this work. The Immigration Service just isn't big enough to do its job."
The BBC's John Manel said the situation meant some suspected individuals might be continuing to work illegally despite being known by the Immigration Service.
He said one example "amongst many" was related by the employment agency ANA in Stoke which recruits for the care and nursing sector.
The firm's managing director Yvonne Thompson told the BBC that it was approached about a woman looking for employment as a care worker who had produced a "really poor forgery" of asylum documents.
Ms Thompson said she telephoned the Immigration Service and was informed the woman was illegal, but it could not do anything about her because of lack of resources.
A managing director of another agency in south-east London told the BBC his company sees suspected illlegal immigrants looking for work every week.
However, he only passed on the paperwork to the Immigration Service every six months because he said nothing seems to happen with the information.
In a statement, the Home Office said it was doubling its enforcement resources over the next three years and was committed to tackling illegal working.
It said the Immigration Service adopted an "intelligence-led approach" to operations.
But it said resources and operations always had to be prioritised to maximise the number of people removed, and for public protection.
'Very bad management'
For the Tories, David Davis said the BBC's revelations were "not a surprise".
"The department is lurching from one target to another," the shadow home secretary said.
"This is a direct outcome of government policy and very bad management."
"This comes as a direction from the highest level, but also, frankly, it is known about by ministers."
Lib Dem leader Sir Menzies Campbell said he was "astonished" at the investigation.
It "undoubtedly reduces confidence in the system", he told the Today programme.
Other people would be encouraged to try to get away with working illegally "if they believe the system is as slack as we've heard", he said.
He called for "a reallocation of resources" by the government so more money and staff were available to tackle the problem.