At about eight o'clock on a dull autumnal morning, a mother is preparing breakfast for her young son in the kitchen of an unassuming private house on a little modern estate in Leicester. The doorbell rings. Outside, a series of people carriers and estate cars are rolling up one by one; out of them tumbling a succession of children in twos and threes, all in traditional Islamic dress.

By 8.30, 26 children – some of them only just old enough for school, some almost grown – are sitting in tight rows on the floor of a little inner room, reciting morning prayers in Arabic and in English. By 9.30, the conservatory has become an infant classroom, the dining room has been taken over by the juniors and in the living room, year 7 and 8 girls are preparing to spread their geography projects across the laminate flooring.

By now, the mother has vanished – she doesn't want her name or address to be used, she says, because already families are turning up at odd hours asking to look round the "school" – and Fatima D'Oyen, director of Manara Education, has taken charge with her small team of staff.

There's no doubting that the Manara academy is a most unusual educational institution. But it's also part of a national trend. Although the number of Islamic schools is still small – around 140 at the latest count, just 12 of them state-funded – it is growing fast. About 60 of these schools have opened in the last 10 years; several in the last couple of months. And the demand from parents seems to be huge – one school in Birmingham recently attracted 1,500 applications for just 60 places. At least five Islamic schools have recently applied to be free schools, although so far only one has been approved.

Manara is one of two Islamic schools that have opened in Leicester this autumn – although in its case, the word "school" can only be used loosely. Manara operates just three mornings a week, and its pupils are registered as home-educated.

Because Manara operates on a part-time basis, it does not need to register with the Department for Education as a school. But the rise in the number of Islamic schools has raised some concerns. Leicester City Council has called for national guidance to ensure that parents who send their children to "flexi schools" like Manara can be sure the staff have criminal record checks and their buildings are safe. And in some areas, full-time schools have opened without registration – meaning that there are no checks on the suitability of their staff or the quality of their curriculum.

D'Oyen aims to open a fully registered, full-time school next year. Until recently, she was the headteacher of another Muslim school in Leicester, but left earlier this year – and decided to start her own school. She quickly found that the formalities required were much more cumbersome than in her native US, where she had previously helped to set up an Islamic school in New Mexico.

"The Department for Education wanted everything done six months in advance; they wanted a plan of the building, they wanted to come and inspect," she says. "They wanted to see our curriculum plans in detail – a lot of rigmarole. And we wanted to be open in September. So legally we are a private tuition service – like a supplementary school, but during the day."

Despite its unconventional setting – D'Oyen was invited to tea with the family who live here and seized on the idea that the house could be turned into a school – the children seem contented and the curriculum varied. Manara is experimenting with Montessori teaching methods, and religious education includes moral and personal discussions as well as study of the Qur'an. The time spent by many children learning the Qur'an at madrasas – often 10 hours a week or more – can rob them of their childhood, D'Oyen believes, and she hopes to provide a more humane alternative. The pupils will learn about gardening and alternative technologies, and have access to the garden, which is used as an outdoor classroom.

"We'd like to teach a long morning, which would include some Islamic education, and then in the afternoons children would have more choice of activities – arts, crafts, PE," D'Oyen says. "We want the children to have creativity in their lives, and to follow some of their interests."

She foresees no problems at all in finding pupils – another Islamic school in Leicester already has five applications for each place. The demand from Muslim parents for an education outside the mainstream is growing, she says.

Others in the Muslim world agree with her. Dr Ghayasuddin Siddiqui, founder of the Muslim Institute thinktank, says there is a growing feeling among Muslim parents that mainstream schools are not serving their children well: "If schools are focused on raising standards and on ensuring that there is discipline, I think most people are happy with that," he says. "But more and more parents are concerned about the quality of education, and about discipline."

Yet in some areas, situations have arisen that have caused concern. A headteacher in the north of England, who asked not to be identified, described how an Islamic school had opened up two years ago without permission opposite her own primary school. "It operated for about six months without registration, and then it was forced to close. It didn't take long before it was registered and reopened again," she says. "Some lovely ladies came to see me and they invited me and my deputy to see what was happening there. But I have to say I found the whole thing very worrying indeed – it's just so divisive." She had been trained as an Ofsted inspector, she said, and did not believe that the school would have been allowed to operate in the state sector. Its buildings, even after renovation, were unsuitable, she said, and its curriculum was too narrow, with every lesson being linked in some way to the Qur'an or the life of the Prophet Muhammad.

The Association of Muslim Schools, set up 20 years ago to support a then-tiny band of institutions, acknowledges that in response to a growing demand for Islamic education, a number of full-time schools have opened without proper formalities.

"The Department for Education is in constant contact with us, and they do tell us if someone's operating without registration," says Shazad Mohammed, the director of the association. "Then we visit to stress the importance of registering – the local authorities have to know where the children are, for safeguarding purposes. We strongly discourage this – it is illegal to operate without registration."

But it is hardly surprising that there should be some breaches, he adds – the UK has two and a half million Muslims, and the number is rising fast. The majority are aged between 13 and 25. One highly regarded Muslim school, the Al-Hijrah school in Birmingham, has introduced a lottery system to allocate places because up to 25 parents are competing for each one.

In Leicester, the city council says it is anticipating a rise in the number of "flexi schools" like Manara, and it has asked the government to address the issue. "It is anticipated that this form of education may become more common, and the local authority has asked that the Department for Education consider producing national guidance for parents and providers around the quality of provision, including criminal record checks, health and safety and planning permission," it said in a statement.

The DfE welcomed Leicester's commitment to working with home educators, but did not respond to requests for a comment on whether there should be more regulation of the sector.

But for Fatima D'Oyen, the road ahead seems clear. Leicester's home education inspector paid her a visit this month, and was apparently impressed. Attempts to regulate the sector further would be counterproductive, she argues. "My perspective is that 95% of parents can be trusted to do what is best for their children," she says. "I don't believe it is either possible or desirable to try to regulate, especially if the desire to do so comes from racism or misplaced paternalism. The reality is that most Muslims setting up or working at Islamic schools, whether part-time, full-time, supplementary or otherwise, do so out of a sense of altruism and wanting to help children get a good education."