"We have never seen anything like it before," said Säpo chief Anders Thornberg in an interview with Swedish news agency TT.
The vast majority of the extremists support violent Islamist ideologies, according to Säpo, whose security experts in a report in 2010 estimated that there were around 200 such sympathizers in Sweden.
"We would say that it has gone from hundreds to thousands now," said Thornberg.
However, he stressed that the security service believes few of them have the ability to, or even intend to, carry out a terror attack in Sweden.
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The security police are working on putting together new, more exact, figures, reports TT. Thornberg described the situation as serious.
"This is the 'new normal' … It is an historic challenge that extremist circles are growing," he said.
He attributed the rise primarily to the propaganda machine of Isis, also known as the Islamic State (IS) or Daesh, which has united different groups of Islamist extremists.
"We used to have different circles. We had radicalized people from North Africa, the Middle East and Somalia, but they were all separate," he said.
Thornberg said Säpo now receives around 6,000 intelligence tips a month concerning terrorism and extremism, compared to an average 2,000 a month in 2012.
Terror expert Magnus Ranstorp of the Swedish Defence University told The Local the new figures were expected. "Those of us working in the field of counter-terrorism are not surprised, it has been in the making for some time, it didn't just appear. When IS declared the caliphate, that was the genie out of the bottle," he said.
"It is a development we're seeing in general, of course it's worse in bigger countries. In the UK there are 23,000 extremists, in Belgium there are 18,000, in France 17,000. Not everyone is equally dangerous, but it only takes a small number of people."
Ranstorp said there were four reasons behind the increase in violent extremists:
"First, there's the mobilization within the context of the conflict in the Middle East, in particular Syria. Then there's social media. There has been a tsunami of carefully calibrated propaganda and recruitment – and with social media IS can reach out to anyone's living room," he told The Local.
A third reason is segregation in Sweden's so-called vulnerable areas, districts with often high poverty and crime rates and a higher prevalence of religious extremism.
"It's important because it has all the socio-economic problems, but also it has been easy for extremists to recruit undisturbed in those areas. And the fourth reason is maybe that the prevention measures have been pretty tame. It's become better in the last year, but I have long said that if you compare Denmark and Sweden, Denmark is at university level and Sweden at kindergarten level," said Ranstorp, calling for sharper prevention measures and emphasizing the need for police and other governmental and local authorities to work together to break segregation in vulnerable areas.
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Interior Minister Anders Ygeman called the figures "worrying", but added: "This is above all a description of what has happened in the past few years and it is already known. We have seen an increasing in violent extremism and an increase in sympathizers too, especially together with the outbreak of the war in Syria."
Asked what he thought the increase in people supporting extremist ideologies said about Sweden's work to combat radicalism, he told TT: "I think it says little. This is a development we have seen in a number of countries in Europe. On the other hand, it shows that it was right to take those measures we have. A permanent centre against violent extremism, that we have increased the budget to work against violent extremism, that we have increased the security police's budget for three years."
Säpo has previously said that about 300 people from Sweden are known to have travelled to Syria and Iraq to join organizations such as Isis since 2012.
Jihadist or sympathizers from Sweden have been linked to several terrorist attacks in recent years.
On April 7th, an Uzbek national who had shown sympathies for jihadist groups including Isis used a stolen truck to mow down pedestrians on a busy shopping street in Stockholm, killing five people and injuring 15.
And a Swedish national, Osama Krayem, has been charged with terrorist murders over the 2016 Brussels metro bombing.
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