The six-week consultation, launched just before Christmas, seeks views on draft guidance from bodies that will be required to follow it.
These include local authorities, early years providers, schools, colleges, universities, the NHS, prisons and the probation service, and the police.
The draft guidance is part of the Government's anti-terrorism strategy, Prevent, published by the Government in 2011.
The Counter-Terrorism and Security Bill, which is currently going through Parliament, will place a duty on early years settings, schools and other institutions to have due regard to the need to prevent people from being drawn into terrorism.
Schools and nurseries have a duty of care to their pupils and staff, the Home Office said.
It added that the new duty to have due regard to the need to prevent people from being drawn into terrorism should be seen in a similar way to settings' existing safeguarding responsibilities.
A Government spokesperson said, 'We are not expecting teachers and nursery workers to carry out unnecessary intrusion into family life, but we do expect them to take action when they observe behaviour of concern.'
Examples given by the Home Office of behaviour that it would not expect to be ignored include if a young child told their teacher that at a religious school, or a madrassah, teaches them that non-Muslims are 'wicked', or if a child made anti-Semitic comments in front of a nursery worker.
The move follows Department for Education (DfE) guidance published in the autumn stating that local authorities must take action to remove early education funding from early years settings that are not promoting British values.
The DfE said that teaching of such values would include learning right from wrong, learning to take turns and share, and challenging negative attitudes and stereotypes. Ofsted will be required to inspect early years providers against these criteria.
Nurseries found to be promoting extremist views will have their early education funding taken away by their local authority. This will include settings that teach creationism as scientific fact.
The Government spokesperson added, 'It is important that children are taught fundamental British values in an age-appropriate way. For children in the early years, this will be about learning right from wrong and in practitioners challenging negative attitudes and stereotypes.
'We are currently consulting on statutory guidance that will set out the types of activity specified authorities should consider when complying with the proposed duty. For schools, including nurseries and other childcare providers, we would expect staff to have the training they need to identify children at risk of radicalisation and know where and how to refer them for further help if necessary.'
Early years organisations urged settings to review their existing policies, but also called for more detailed guidance on implementing the plans.
Michael Freeston, director of quality improvement at the Pre-School Learning Alliance, said, 'It's important that practitioners take a proportionate and considered response to what is still an emerging issue.
'As a first step, providers should review their existing equality, behaviour and safeguarding policies and procedures, as well as their terms and conditions to ensure that both staff and parents are clear on how any relevant situation - such as a child making comments deemed to be inappropriate or concerning - would be dealt with. Such an approach will help ensure the maintenance of early years environments that consistently promote positive values of equality, tolerance and respect for others.'
Purnima Tanuku, chief executive of the National Day Nurseries Association, said, 'The Prevent consultation covers some very serious new responsibilities for nurseries, yet there is no specific guidance for early years settings on how to fulfil these.
'Nurseries will be required to identify children at risk and challenge extremist ideas, and ensure staff are trained to do so. Any action taken by nurseries to exercise these duties must be proportionate and age appropriate.
'In our consultation response, we will be pressing for detailed guidance to early years providers and support for training and implementation.'
The consultation closes on 30 January.
- Find Prevent Duty Guidance: a consultation at www.gov.uk/government/consultations/prevent-duty.