David Cameron has revealed he and wife Samantha are considering ‘three or four’ state secondary schools for daughter Nancy, insisting no one should feel the need to pay for private education.

The Prime Minister, who had a privileged schooling at Eton College, confirmed he is set to become the first Conservative prime minister to send a child to a state secondary. ‘We want to, I have always said that. We have been round three or four schools in London and had a good look,’ he said.‘My daughter has got a very large say in the matter so she is having a think about what she has seen. But there are some good schools to go to.’ The Camerons are understood to have looked at The Grey Coat Hospital Church of England School, an all-girls comprehensive three-quarters of a mile from Downing Street, for 10-year-old Nancy. The Government’s chief whip and former Education Secretary Michael Gove has sent his daughter Beatrice, who attended the same west London church state primary as the Cameron children, to the school.

Mr Cameron and his wife have also visited Holland Park School – one of the first comprehensives in the UK, where Labour grandees such as the late Tony Benn sent their children. Both schools are described as 'ethnically diverse' and have intakes from some of London's most deprived areas. Mrs Cameron - who was educated at Marlborough College – is said to have told friends she is keen for Nancy to have a 'normal education'.

Mr Cameron, in an interview with Good Housekeeping magazine, suggested the Government’s education reforms were designed to make private education redundant. ‘If you pay your taxes you shouldn't have to pay all over again. There is no reason why our state schools can't be among the best in the world, and some of them are,’ he said.

'What is exciting is there this change not only in practice but also in culture which is all about excellence and wanting to be the best and wanting to get the best out of every child, and you are now seeing that in more and more schools.' The Prime Minister added: ‘It’s about early intervention, intolerance of failure and getting stuck in. We know what works now; it is changing the leadership, changing the curriculum, better discipline, uniforms.

We can have quite a quick effect. ‘The proof of the pudding is that since 2010 there are a quarter of a million fewer children in failing schools – that is quite a lot of schools. This is recognising that in some areas schools slide into failure and not enough happens quickly enough and that is where the regional schools commissioners who tend to be former expert head teachers can get stuck in. ‘There are endless debates about how you arrange admissions. Should we have lotteries? Should we have selections? In the end, the answer to all these questions is let’s have more good schools. We are introducing more good schools and more good school places through school turnarounds, academies, the free schools. We need that to go further and faster.’ The Prime Minister rejected the suggestion that the Government’s exam reforms, which less emphasis in coursework, meant less academic children would lose out.

‘I don’t agree with that. I think all children will benefit because I think all children deserve to have the fundamentals of a good education. I simply do not accept there are some children who can’t get a good qualification in English or Maths,’ he said. 'I say to my own children “there isn’t a job in the world where you don’t need a qualification in English or Maths”. My son said “what about being a footballer?” I said “you still have to read the contract and count the money”.' Asked how often he sits down with his children and does homework with them, Mr Cameron said: ‘In a week I will do some help, but Samantha plays the leading part, I will admit. This morning, I was trying to get one to do guitar practise in one room and the other to do touch typing in the other. They were running me ragged because they know that when I am in charge it is not quite so well organised. ‘I got there in the end, various threats were … Screen time [on tablets] in the Cameron household is severely rationed to try and deliver the effect of homework, but it doesn’t always work.’ Mr Cameron said he was most inspired at school by Michael Kidson, one of his history teachers, now 84.

‘He was brilliant but used teaching methods that you would not necessarily get away with now. He had a wooden block on his desk and if you weren’t concentrating he would fling it across the room at you,’ the Prime Minister said.

Education Secretary Nicky Morgan told the magazine she had also chosen a state primary for her son. 'I think you have more good and outstanding [state] schools. The choice is there. My son is at a good, local village school and there was really no question that was where he went after nursery,’ she said.

A Number Ten spokesman said: 'As with other parents at this time of year, the Prime Minister and Mrs Cameron are looking at various schools for their daughter to go to next September.'