The education secretary, Nicky Morgan, has said it is “very much up to schools” if they want to ban the full-face veil, arguing that being able to see a person’s mouth was important for teaching.

Morgan was speaking after the chief schools inspector, Sir Michael Wilshaw, said inspectors had found that religious dress that covered the face sometimes caused communication problems.

David Cameron had said on Monday that he would back institutions with “sensible rules” over Muslims wearing full-face veils, but he ruled out a full public ban.

Speaking to BBC Radio 4’s Today programme on Tuesday, Morgan said: “The prime minister was absolutely right to say, and we have a very clear view in this country, we are not going to tell people what they can and can’t wear; that would cut across the values we are talking about that we want everybody to follow.

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“But there are times, there are institutions and organisations, where it is right – schools will be one of them – where the school leaders want to have a clear uniform policy they want everybody to observe and they may decide that point, that they don’t want [pupils or teachers] to wear the full-face veil.”

She added: “It very much is up to the schools; schools will have a uniform policy. But there are certain things, particularly in relation to teachers who are teaching young children, particularly learning to read and to speak, where actually seeing the teacher’s mouth is very, very important in understanding them.”

Morgan made the comments ahead of a speech at London’s Bethnal Green Academy, which was attended by three girls who ran away to Syria, on the day that she launched a government website providing advice for parents who are worried their children could run away to join Islamic State.

Wilshaw told BBC2’s Newsnight on Monday that the veil was “possibly” stopping teachers and pupils from communicating well and that he backed individual schools choosing to stop Muslim girls wearing the veil, “particularly if it is stopping good communication in the classroom and in the lecture hall”.

“My inspectors say on occasions they go into classrooms where they see there are problems about communications,” he said.

His comments came after Cameron announced that migrants who failed language tests after two and a half years in the UK might be forced to leave.

“The prime minister’s view that we have got to make sure that our liberal values, our liberal west values, are protected, people need to listen to that,” said Wilshaw, who is head of Ofsted.

“The Muslim community needs to listen to it as well. We have come a long way in our society to ensure that we have equality for women and that they are treated fairly. We mustn’t go backwards.”

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Speaking on Monday morning, promoting his plans to encourage greater integration of Muslim women in the UK, Cameron ruled out the idea of imposing a French-style ban on full-face veils in public, but said he would support institutions that needed to “see someone’s face”.

“In our country people should be free to wear what they like,” he told the Today programme, adding: “When you are coming into contact either with different institutions or, for instance, you are in court, or you need to see someone’s face at the border, then I would always back the authorities or the institutions that have put in place proper and sensible rules.

“Going for the French approach of banning an item of clothing, I do not think that’s the way we do things in this country and I do not think that would help.”