Education Secretary Nicky Morgan was also today to announce plans to force schools to help stop teenagers travelling abroad to fight alongside jihadist groups such as Isis.
Schools will be required to inform councils when pupils stop attending without any explanation and Muslim parents will be encouraged to carry out checks to ensure their children are not being radicalised.
Cameron has also announced that tens of thousands of Muslim women will face deportation unless they pass a series of English language tests after coming to Britain on spouse visas.
The Prime Minister's comments about veils will reignite the row over whether British institutions should be able to stop women covering their faces for religious reasons in public places.
He refused to endorse a French-style blanket ban but made it clear that individual organisations can choose to stop Muslim women wearing the veil.
The Daily Telegraph in 2013 disclosed that more than a dozen National Health Service hospitals had told staff not to wear the niqab - a full veil which covers the face - while in contact with patients.
The same year, a London judge ordered a Muslim defendant to remove her veil, but asked politicians for clearer instructions on veils in court. Several Conservative MPs want the Government to consider a full ban on the veil.
"I think in our country people should be free to wear what they like, within limits live how they like, and all the rest of it," Cameron told BBC Radio 4.
"What does matter is if, for instance, a school has a uniform policy, sensitively put in place and all the rest of it, and people want to flout that uniform policy, often for reasons that aren't connected to religion, you should always come down on the side of the school.
"When you are coming into contact with an institution or you're in court, or if you need to be able to see someone's face at the border, then I will always back the authority ... that [has] put in place proper and sensible rules. Of France's 2010 ban of full-face veils, he said: "I don't think that's the way we do things in this country and I don't think that would help."
Conservative MP Philip Hollobone welcomed the comments as a step in the right direction, "but given the stridency with which Muslim groups advance their cause, sooner or later this will be put to the test", he said.
"It should apply to any public official including schools, hospitals, councils, the police, border force, hospitals, GP surgeries ... I don't want to live in a country where a police officer is veiled, where a news reader is veiled, where a nurse or doctor is veiled."
The Telegraph also understands ministers are drawing up guidance to ban gender segregation in public meetings in public-owned buildings.
It follows concerns that Muslim men and women were separated in a series of meetings in the run up to the election.
Morgan was to give a speech at the Bethnal Green Academy, east London, which came to public attention last year when four of its pupils fled to Syria to become "jihadi brides".
She was to announce a new website to help parents and teachers identify potential victims of radicalisation.
It warns if "excessive time spent online or on mobile phones" as well as a "susceptibility to conspiracy theories and a feeling of persecution".
She was also to announce plans to ensure schools register with authorities any pupils that stop attending lessons.
Cameron announced yesterday that immigrants will have to show they have improved their English after two-and-a-half years or face being deported.
Under the plans tens of thousands of migrants who come to Britain on a "spouse visa" to join their husband or wife will have to pass an additional English language test after two years.
Naz Shah, a Muslim Labour MP for Bradford West, backed his call for more women to learn English.
-Telegraph Group Ltd
