Theresa May, the home secretary, has proposed a raft of measures to combat the radicalisation of Muslims, including new controls on the internet and the banning of groups preaching hate.

Under current terror legislation, home secretaries have the power to proscribe groups suspected of being "concerned with terrorism", and actions in support of that group can lead to prosecution.

May also confirmed that she was still fighting to ensure the full communications data bill – the so-called snooper's charter – is passed by parliament. She has won the support of the former Labour home secretary Alan Johnson, who said she should resign if the cabinet was not prepared to back her.

Johnson's remarks were aimed at the deputy prime minister, Nick Clegg, who has said he will allow only some small measures through.

May said she believed that the budget of the counter-terrorism police had to be treated differently from the rest of the police force in the current spending review and that it should be fully protected, as it was in the 2010 spending review.

May said: "I'm absolutely clear that we need to ensure that the intelligence services and, indeed, in policing CT (counter-terrorism policing) … in the last spending review we ensured that CT policing was not treated the same as overall policing and I see every reason to take that same view in the next spending review."

May said the government review of its anti-terror strategy needed to look "at the question of whether perhaps we need to have banning orders to ban organisations that don't meet the threshold for proscription".

May questioned why the BBC had allowed a radical cleric, Anjem Choudray, to appear on Newsnight last week. "I think Anjem Choudary has disgusting views and I think it is right that we look at how those views are being presented and I think there were many people who did indeed say what is the BBC doing interviewing Anjem Choudary?" she said.

But May did not criticise TV organisations for allowing video clips of those claiming to have been behind the Woolwich murder to be broadcast in the immediate wake of Drummer Lee Rigby's killing.

She also confirmed that the government's counter-terror strategy, Prevent, would be reviewed for a second time since the government came to power.

The first review led to a focus away from violent extremists to extremism in general, but it is clear that the government will again put pressure on universities to tilt the balance away from free speech to clamping down on the teaching of extremism.

The former communities secretary Hazel Blears criticised the government for cutting the Prevent budget. She said that it was focusing too much on those already on the edge of extremism, rather than trying to forestall a broader culture of extremism.

May said new rules from Ofcom had led to more than 5,500 unacceptable "terrorist" messages preaching violence or jihad being taken down from the internet. But she said she wanted to see what more could be done to police the messages circulated online.

She said: "There is no doubt that people are able to watch things through the internet which can lead to radicalisation. What we do is we have a referral unit which members of the public and organisations are able to refer into that unit when they've got concerns about what's being broadcast across the internet.

"There has been discussion of a greater use of court orders to block some sites, but it will be difficult to decide whether responsibility will lie with the Home Office or internet service providers."

Johnson said he was a passionate supporter of the communications data bill, insisting this was not a knee-jerk reaction to the Woolwich killing.

He said: "For six years at least, we've known that as telecommunications has moved on, and sophistication has moved on, the security forces who need to know – not to open these communications, that's a completely different process, it's nothing to do with this – it's to know who's been contacting who.

"I am absolutely with the home secretary on [that] … I'm absolutely confident she'll be fighting for this. We need to get this on the statute book before the next general election. And I think it's absolutely crucial. Indeed, I think it's a resignation issue for a home secretary if the cabinet do not support her in this central part of what the security services do."

Lord Howard, the former Conservative leader, claimed the Tories could form a parliamentary alliance with Labour to override Liberal Democrat objections and force through the bill.

In reality Labour's official position is less enthusiastic about the legislation than Johnson. The shadow home secretary, Yvette Cooper, has trod a fine line, saying security services may need greater powers but they must be proportionate and subject to checks.

Menzies Campbell, the former Liberal Democrat leader and member of the intelligence and security committee, supported Clegg, saying the proposals in the bill represented a major intrusion into people's lives and would allow the security services access to every phone call, text and internet site visited.

He said: "It would be the first time such powers could be given to monitor people at risk of being radicalised. We would be using electronic listening and other devices, as I understand it, for the first time for people who were not suspected of indulging in criminal behaviour.

"What is being argued for by some is that all of us should be subject to scrutiny of every telephone call we make, every text we send and every internet site we visit. That is a very, very substantial intrusion into people's lives."

He said the security services could undertake this surveillance at present if they sought a warrant in court. He said it would be wise for David Cameron to put the proposals under review through the proposed Downing Street taskforce.