Leaden Hall Prep School in Salisbury (stock picture)Credit:Philip Hollis Want the best of The Telegraph direct to your email and WhatsApp? Sign up to our free twice-daily Front Page newsletter and new audio briefings. In September Prime Minister Theresa May warned Britain’s independent schools are “divorced from normal life” and said they must do more to help children from poorer backgrounds or face losing their status as charities.She was accused of “waging war” on private schools by announcing a series of new policy measures which threatened them with punishment unless they complied.However, prep schools have claimed that poor parents are being put off by negative stereotypes of private schools that portrayed them as elitist and quirky.A survey of 200 prep school heads was conducted by Attain, the magazine of the Independent Association of Prep Schools, to seek their views on Mrs May’s plans. Of these, 125 said their schools did offer full bursaries.However, 9.6 per cent of head teachers said they had “no interest” from parents and had not offered any full bursary places in the past year, and 39.2 per cent said there was “not enough interest” and their schools rarely got enough suitable candidates.Another 28.8 per cent said there was “sufficient interest” and they usually allocated all their free places while only 22.4 per cent said they had “lots of interest” and demand outstrips supply.In contrast, 80 per cent of prep school heads said they were able to allocate part-funded bursaries for families who can afford to pay a proportion of the fees but not the full amount. Average fees are £13,623 a year for day pupils and £30,951 for boarding.Private schools offered 39,825 means-tested places to families who could not afford full fees last year, 7.7 per cent of all pupils at independent schools, with an average reduction of £8,370.Earlier this month it emerged independent schools will tell the government they are prepared to offer 10,000 free places to children from low-income backgrounds. The Independent Schools Council (ISC) said private schools in England will cover the costs each year if the Government gives them the £5,550 per pupil the state system allocates for each child.David Hanson, chief executive of the Independent Association of Prep Schools, said: “The media characterisation of private schools is so extreme and embedded through constant repetition that for ordinary people what they represent is not only unattainable, but also incomprehensible and alien.”I welcome the opportunity we have now to bust those myths.”Ryan Shorthouse, director of the progressive Tory think tank Bright Blue, said: “It is irksome to see some of these prestigious and well-resourced institutions now complain that the government is asking them to do activities sponsor a state school, or offer fully funded bursaries which they allege they are unable to do.”They should try harder in making sure they can meet these reasonable requests, or start suggesting reasonable alternatives.” Free private school places set aside for children from low-income backgrounds are going unfilled because heads cannot find eligible families to take them, it emerged today.Researchers found almost half of independent prep schools said they did not get enough applicants for fully funded bursaries.One in ten said they had not offered a single free place in the past year.It comes as private schools face a threat to their charitable status, which the ISC says saves such establishments £150 million per year.