These threats are being made to avoid dismissal or to win promotion - and many companies choose to knuckle under quietly rather than risk the damaging publicity of such a claim, says Stephanie Klass, an employment law expert for the Berg firm of solicitors in Manchester.
This scenario, which is backed up by other lawyers working in the field, comes as analysts in the United States are expressing doubts about the efficacy of policies that are intended to combat sexual harassment.
Some suggest that the practice of measuring compliance with sexual harassment laws by the zeal with which incidents are prosecuted is creating the wrong impression. Although a lack of investigations may suggest that a company is soft on discrimination and harassment, it may equally simply indicate that the organisation has established a comfortable working environment for its staff.
The problem, as Mrs Klass sees it, is that such incidents come down to one person's word against another's. The company that does not believe the claims of a female employee risks being branded discriminatory and being attacked for not taking women seriously. At the same time, not believing the man can end his career.
"I have no wish to overplay this," Ms Klass says. "It's a minority. But it's a minority worth thinking about." She stresses that she is not attacking the system, just pointing out that it can be exploited.
In stark contrast to the traditional view of sexual harassment involving bosses chasing secretaries, more women than ever before are prepared to make false accusations against senior male colleagues because they know that many companies are not prepared to fight the claim before an industrial tribunal, according to Mrs Klass.
"The number of such cases is growing, and I am dealing with more than ever before."
Her most recent case involved a company chairman who was falsely accused of serious sexual harassment by a junior employee he had just fired. The allegations were dismissed by the tribunal, but the businessman concerned was seriously affected by the experience.
The problem had arisen because the man decided to sack the woman during an individual meeting, because he knew her personally and thought it would be the best way of dealing with the matter. That meant, of course, that there was nobody present to corroborate his story.
The answer, says Mrs Klass, is to be "circumspect about what you do - as with any aspect of employment law".
She says she suspects that the number of cases of this type that reach tribunals represents only the tip of the iceberg. "Many companies pay up because they know an embittered employee can make allegations without any proof."
She advises companies faced with such false claims to seek the best legal advice. "They should always think twice before bowing to blackmail of this kind."