He added: "There would need to be a continuous assessment of those people to find out their psychological state. I agree, however, that they are often the best people to speak to young people, in the same way that former gang members do. They have the credibility and can be more emotional about it.
"They can explain their own experiences better to those potentially vulnerable young people - and why they have changed their minds."
Under the new Counter-Terrorism and Security Act and the Prevent anti-terrorism programme, school staff are already required to look out for signs of radicalisation among students.
But this week the pressure on teachers intensified after it emerged that Talha Asmal, a 17-year-old pupil from Mirfield Free Grammar and Sixth Form in Dewsbury, West Yorkshire, had died after reportedly detonating a vehicle fitted with explosives while fighting for IS in Iraq.
Paula Sherriff, Labour MP for Dewsbury, called on schools to work with "mosques, churches, community groups and parents" to ensure that "no more impressionable young people are brainwashed by those behind Isis".
Talha Asmal's parents said that their son had been groomed online by extremists.
Speaking at an event organised by the Varkey Foundation, the philanthropic arm of the GEMS chain of private schools, and the Club de Madrid organisation of former world leaders, Professor Neumann said that more needed to be done to combat the messages that were getting through to the country's young Muslim population via the internet.
He said the world was "witnessing the greatest mobilisation of jihadists in memory" in a movement that included between 600 and 700 recruits from the UK. "For the first time ever we're seeing exceptionally young people going [to fight for terrorist organisations]," he said.
Track and trace
Professor Neumann added that his department was able to track and trace jihadists who had travelled to join IS via social media, and had even made contact with them on Facebook. Often, the young people spoke of their disillusionment once they arrived in places such as Syria, he said.
In February, three schoolgirls from Bethnal Green Academy, Tower Hamlets, went missing and were later found to have boarded a flight to Turkey during half-term. The group - Amira Abase, 15, Shamima Begum, 15, and Kadiza Sultana, 16 - are believed to have crossed the border into IS-controlled Syria to become so-called "jihadi brides".
The girls' disappearance follows that of another 15-year-old pupil from Bethnal Green Academy, Sharmeena Begum, who went missing in December after travelling to Syria via Turkey.
Robert McCulloch-Graham, director of education at Tower Hamlets, said any attempts to bring former extremists into the classroom would require serious "checks and balances", but added that such debate was necessary.
"The natural reaction among schools, particularly where there's been a major event, such as at Bethnal Green Academy, is to close everything," he said. "Some schools had an Islamic Society and they shut them down in response to what has gone on.
"But, actually, what should be happening is the opposite. We should be stimulating debate in these schools."
`A really good idea'
Graham Best is principal of St John's College. The independent school in Portsmouth was attended by Mehdi Hassan up to the age of 14, before he joined Islamic State aged 19. The young man, whom Mr Best describes as "quiet and studious", was killed while fighting in Syria.
Mr Best says that as long as all the precautions are taken, he thinks the idea of sending former extremists into schools is a "really good one", but it depends on the person.
"They can't be seen to be glorifying it, but they also can't go the other way as to turn people off from listening," he says.
"They would also need to be able to deal with some challenging questions as there would be some differing views from the students."