A Department for Education consultation document expects schools to confront intolerant or extremist views among staff, parents and pupils.
It follows allegations of hardline Muslim takeovers in Birmingham schools.
Education Secretary Michael Gove wants all schools to promote British values.
Mr Gove’s comments followed Ofsted’s conclusion that “a culture of fear and intimidation had taken grip” in Birmingham schools caught up in the Trojan Horse claims.
The watchdog found evidence of an “organised campaign to target certain schools” with head teachers marginalised or forced out of their jobs.
Five of 21 schools inspected were put into special measures, with a sixth labelled inadequate for poor educational standards.
Independent schools, academies and free schools already have to adhere to the Independent School Standards (ISS) which demand that schools encourage pupils to “respect” British values.
Ofsted is planning to introduce an equivalent for local-authority-run state schools later this year.
From September, the ISS will be tightened up under the proposals, with “a stronger basis for swift intervention in schools which are not actively promoting British values”.
The proposals do not alter the definition of British values, first set out in the UK-wide 2011 Prevent Strategy and part of the ISS since January last year.
These include democracy, the rule of law, individual liberty, mutual respect and tolerance of those of different faiths and beliefs.
The plan tightens up the rules on what schools are free to teach in terms of politics and religion.
Under the changes, any teaching of extremism would be banned as would the promotion of partisan political views.
Democracy and law-making
Schools would be expected to promote respect for their own and other cultures as well as respect for public institutions and services in England including democracy and law-making.
Students will also be expected to understand how they can contribute to wider society.
Schools will also be expected to challenge pupils, staff or parents expressing opinions contrary to fundamental British values.
A Department for Education spokesman said: “Keeping our children safe and ensuring schools prepare them for life in modern Britain could not be more important.
“This change is an important step towards ensuring we have a strong legal basis for intervening in those schools where this is an issue.
“The vast majority of schools already promote British values – this is about making sure we have the tools we need to intervene if children are being let down.”
The consultation closes in August.