A top level taskforce is to be set up in the wake of the killing of a soldier in Woolwich to “look again” at the government’s strategy for dealing with radicalisation and extremism.
The special committee, to be chaired by the prime minister, will include senior cabinet ministers and security chiefs.
Downing Street sources say it will “build on” Labour’s Prevent Strategy.
Police probing Lee Rigby’s death have arrested three more men on suspicion of conspiracy to commit murder.
Michael Adebolajo, 28, and Michael Adebowale, 22, were arrested at the scene of Wednesday’s killing of Drummer Rigby, 25, near Woolwich Barracks, and remain in hospital after they were shot by police.
Shortly after the attack, Mr Adebolajo was filmed by a passer-by saying he had carried it out because British soldiers killed Muslims every day.
Meanwhile, French authorities are investigating whether the stabbing of a soldier in a Paris suburb was a copycat attack in the wake of the murder of Drummer Rigby.
The coalition government reviewed Prevent in 2011 – set up after the 2005 London transport network bombings – and said it had uncovered serious failings with the existing policy.
The home secretary said at the time that some of the money was going to groups who should have been confronted, not given funds.
Theresa May also said the relaunched policy should recognise the impact of non-violent extremism.
The coalition reduced the annual budget for Prevent from £63m to £36m.
After the events in Woolwich the prime minister and some around him believe the whole approach on dealing with Islamist extremism needs an informal review.
Outlining the new taskforce, the Number 10 sources told the BBC it would be “foolish not to look again at how things are being dealt with”.
The government said it would consider blocking some violent and unlawful websites, but the Home Office has confirmed that is still in the developmental stage.
Downing Street wants to look at how serious the scale of radicalisation is, something it says involves “hundreds of individuals”.
The prime minister also wants to focus on the range of powers available for dealing with extremist preachers or figures who are “fostering extremism”.
Sources say work being done to counter radicalisation in places such as universities and prisons needs to be “maximised”.
Labour sources say it is “right to develop all these areas” but they say much of this was promised after the review two years ago.
They said the taskforce needed to look at “what hasn’t worked” since then, “not re-invent the wheel”.
The Labour chairman of the Commons Home Affairs Select Committee has criticised the government for what he says are serious cuts to funding for frontline initiatives, which engage with vulnerable young men.
Keith Vaz told the BBC that the coalition should change the name of its strategy, saying “don’t call it Prevent, call it Engage”.
He also said the government needed to take firmer action against internet search engines who give access to some of the material used to radicalise people, because, he said, “It’s not happening in universities or schools, but in bedrooms.”
Meanwhile, a senior backbench Conservative MP has warned against a high profile over reaction.
Bernard Jenkin, who chairs parliament’s all-party group on security, said: “I suspect it will prove more effective in the long run just to get on with this in the background… the underlying message should be, ‘Keep calm and carry on.'”
Scotland Yard said the latest arrests were made in south-east London by detectives from the Counter Terrorism Command supported by specialist firearms officers.
A 24-year-old man and 28-year-old man were arrested at a residential address while a 21-year-old man was held in the street in Charlton Lane, Charlton.
A Taser was used on the 28-year-old man and the 21-year-old but the suspects did not require hospital treatment.
Police said they were also searching four residential addresses in south-east London.
Scotland Yard added that a 29-year-old man arrested on suspicion of conspiracy to murder has been bailed to return pending further inquiries.