The Home Secretary has criticised universities for their "complacency" in tackling radicalisation and Islamic extremism on campus.
The remarks come ahead of the publication of the Government's revised Prevent strategy, which was originally launched in 2007 to stop the growth of home-grown terrorism.
In an interview with the Daily Telegraph, Theresa May said: "I think for too long there's been complacency around universities. I don't think they have been sufficiently willing to recognise what can be happening on their campuses and the radicalisation that can take place. I think there is more that universities can do."
The Government has identified 40 English universities where there could be a "particular risk" or radicalisation or recruitment on campus, according to the Daily Mail which said it had seen the updated strategy.
"More than 30% of people convicted for al Qaida-associated terrorist offences in the UK... are known to have attended university or a higher education institution," the newspaper quoted the report as saying.
It is understood the document also raises concerns over the Federation of Student Islamic Societies and what is seen as an insufficient willingness to tackle extremism.
Mrs May told the Telegraph: "They need to be prepared to stand up and say that organisations that are extreme or support extremism or have extremist speakers should not be part of their grouping."
The newspaper said the strategy would contain details of partnerships with YouTube and AOL aimed at combating extremism online, as well as moves to limit access to extremist websites from schools and public libraries.
The report will also name the 25 boroughs most at risk from Islamist extremism, including areas of London, Birmingham, Leeds, Bradford and Manchester, it was reported.
It is understood about 20 of the organisations which have received funding over the last three years will have their funding cut.
Mrs May said: "It's a result of a close look at the values of the organisations themselves."
Mrs May launched the review of the Prevent strategy - overseen by the independent counter-terrorism reviewer Lord Carlile of Berriew - last November saying it was not working as well as it could be.
The £60 million Government programme was adopted in the wake of the July 7 bombings, and aims to counter militant Islamism by supporting mainstream Muslim groups which offer an alternative to extremism.
But Prevent was criticised by some Muslims who said they feared it was being used to spy on their communities, and by other ethnic groups who believed they were missing out on financial support by comparison.
A Home Office spokesman said: "The Government is currently reviewing the Prevent programme, which isn't working as well as it could.
"We need a strategy that is effective and properly focused. The findings will be published shortly."
Among those arrested for terrorism offences who have been linked to British universities is Umar Farouk Abdulmutallab, the so-called "underpants bomber".
He was detained on Christmas Day 2009 accused of trying to blow up a flight from Amsterdam to Detroit.
He graduated a year and a half earlier from University College London, where he was also president of the student Islamic Society.