A crackdown on Islamist extremism could "undermine" the values the Government aims to defend, according to the police chief leading the UK's anti-radicalisation scheme.

Sir Peter Fahy warned that David Cameron's vow to "aggressively" pursue and disrupt extremists trying to spread the "poisonous ideology" of radical Islamism could "backfire".

The chief constable of Greater Manchester Police, who will retire from the service this month, said the strategy could turn officers into "thought police" and call into question what the Government has termed the "British values" of democracy, equality and tolerance.

Sir Peter, national policing lead for the Prevent anti-radicalisation strategy, told the Guardian: "There is a concern that efforts to control extremist narratives will limit free speech and backfire if we don't get the balance right. The efforts to control extremism and limit protest ... may undermine the very rights and British values you seek to protect."

His intervention comes after the Prime Minister, launching new counter-extremism measures, warned it was no time for "kid gloves".

The plan includes a review of the rules of British citizenship, including strengthening the "good character" consideration and whether an applicant has promoted extremist views, while there will be bans on radical preachers posting material online, and internet firms will work more closely with police to stop extremist material being disseminated.

Sir Peter voiced concern over how the new powers would be policed.

He added: "It draws the police in(to) areas the public will be uncomfortable with if they feel that it erodes free speech or religious freedom or the right to protest. At what point do you erode the British values you are trying to protect. Such as live and let live, and freedom of speech.

"The challenge is where do you draw the line on that crossing point and that is what the consultation process needs to address."

The Government's latest strategy document emphasised the need to root out extremism across a wide range of institutions including schools, universities, local councils, the National Health Service, charities and prisons.

There will be an extension of powers for passports to be withdrawn from young people at risk of travelling abroad to join groups such as Islamic State (IS).

The mechanism has already been used on several under-16s since it was introduced in July, and will now be available for 16 and 17 year olds.

Anyone with a conviction or civil order for terrorist or extremist activity will also be automatically barred from working with children and vulnerable people.

The Muslim Council of Britain warned that the strategy was "flawed" and counter-productive, and alienated the Muslim communities whose support the Government was seeking.