Police in Birmingham are investigating whether a document outlining an alleged Islamic fundamentalist plot to "take over" schools in the city was a hoax connected to an employment tribunal involving one of the schools named in the plot.

Detectives have reopened a fraud investigation centred on Adderley primary school in the Saltley area of Birmingham following widespread media publicity of a four-page dossier outlining what the unknown authors called Operation Trojan Horse.

A source at West Midlands police said one line of inquiry is whether the Trojan Horse document is a hoax, linked to other claims of fraud brought by four former teaching assistants at Adderley primary.

Operation Trojan Horse purported to be a "how to" guide for hardline Islamists wanting to commit jihad by overthrowing headteachers at state schools in Birmingham and beyond, and replacing them with compliant leaders who would run the schools on "strict Islamic principles".

Adderley was one of five schools cited in the dossier as a target for infiltration by Islamist hardliners. But the dossier contains several factual errors, casting further doubt on its authenticity.

The teaching assistants – three Muslim women of Pakistani origin and one white, non-Muslim woman – are suing the school for unfair dismissal, claiming they were ousted after someone faked resignation letters carrying their signatures.

Their employment tribunal was due to begin on Monday this week but was postponed by a judge after West Midlands police decided to look again at allegations from the four claimants. In January 2013 they told police they had been forced out of the school in December 2012 as a result of the "fake" letters.

They say they told Adderley they had no intention of quitting and that their signatures had been forged after they raised grievances about their treatment at the school with the local authority.

It is thought that detectives, as well as the council, are examining whether the Trojan Horse plot could have been concocted to support Adderley school's case at tribunal. All four teaching assistants are mentioned in the dossier, albeit not by name, as being part of a plot to oust Adderley's headteacher, Rizvana Darr.

She is described in the document as being "not a good Muslim" who was "not open to our suggestions of adhering to strict Muslim guidelines". Darr was highly praised in the school's last Ofsted report, which graded Adderley "good" overall, with "outstanding" leadership and management.

The alleged plotters note that she is "procedurally strong and so we had to find a reason for her to be sacked linked to procedures". The document says "three of our Muslim sisters ... along with an English woman who is their close friend, have raised an allegation of fraudulent resignation letters against the head (even though they did actually write the letters themselves)".

But a legal source said a handwriting expert had provided a forensic report for the tribunal that strongly suggests none of the teaching assistants signed their own resignation letters.

It is understood that Birmingham city council advised Adderley's leadership team not to fight the tribunal without properly investigating the fraud allegations. The council withdrew the usual indemnity granted to council-run schools in the event of legal action because they thought the risks were too high, fearing reputational damage to the council and the school.

Two parents at Adderley said they had contacted their local MP, Liam Byrne, to complain about the school, and it is understood the local authority is carrying out an investigation into their complaints.

But they deny demanding any of the measures suggested in the Operation Trojan Horse document, such as segregated swimming or a ban on sex education.

One said she had asked the school to let boys and girls get changed for PE separately after a boy asked her daughter to "show him her bum", but said it had nothing to do with religion, just privacy. Another complained her son had been banned from breakfast and after-school clubs without explanation.

The school and head declined to comment or respond to a list of questions sent by the Guardian on Thursday asking about a potential link between the employment tribunal and the Trojan Horse letter.

West Midlands police said they thought the Trojan Horse letter was not a criminal matter but warranted further investigation by the city council and the Department for Education. The force said its investigation into allegations of fraud at Adderley had been reopened by the economic crime unit as a result of the letter coming to light.

Hilary Owens, one of the four teaching assistants who is fighting Adderley at the employment tribunal, welcomed the re-opening of the police investigation.

Her solicitor, Daniel Zakis, from Wallace Robinson and Morgan solicitors, said on Wednesday: "Whilst she [Owens] is disappointed by the decision to adjourn the tribunal she remains steadfast in her desire to pursue the matter. She is confident that the police investigation will support her position and we remain instructed to pursue her claim vigorously."

Other aspects of the dossier point to it being a hoax. It misnames senior officials at both Birmingham and Bradford councils and makes a misleading reference to Springfield school in the Moseley area of Birmingham. The alleged plotters refer to a headteacher called Noshaba Hussain, whom they claim to have ousted from Springfield, only for her to be reinstated by the governors and so "we have another plan in place to get her out". But Springfield confirmed on Thursday that Hussain left the school 20 years ago.