UK faces two more years of severe terror threat, Home Office warns
Security agencies are also confronting a rising risk from extreme right-wing violence as the potential sources of attacks becomes more diverse.
The assessments emerged as the Government prepares to unveil a strengthened counter-terrorism strategy.
Powers were reviewed after five attacks last year, and on Monday Home Secretary Sajid Javid will announce a range of steps aimed at boosting the authorities' efforts to stop atrocities.
Plans to share information held by MI5 more widely across Government and local agencies are expected to be included in the blueprint.
The Home Office said: "In summary, we expect the threat from Islamist terrorism to remain at its current, heightened level for at least the next two years, and that it may increase further.
"We assess the threat from extreme right-wing terrorism is growing.
"Globally, terrorist groups and networks of all ideologies continue to develop organically, exploiting social media, technology and science to further their aims and ambitions."
Security agencies and counter-terror policing have foiled 12 Islamist and four extreme right-wing plots since March last year.
MI5 and police are running more than 500 live operations involving roughly 3,000 "subjects of interest" at any one time.
In addition, there are in excess of 20,000 people who have previously been investigated and who could again pose a threat.
Security chiefs are particularly concerned about the potential risk of individuals in the larger group being rapidly radicalised to the point of violence before the shift is detected.
Salman Abedi, the Manchester bomber, was categorised as a "closed subject of interest" at the time of his attack.
The strategy launched on Monday is expected to set out plans for MI5 to share its intelligence more widely and work with partners such as local authorities on how best to manage the risk posed by closed subjects of interest.
It emerged last year that MI5 had made a commitment to allow knowledge derived from intelligence to be shared more widely beyond intelligence circles.
An official review into four of the five attacks in 2017 by former terror laws watchdog David Anderson QC said: "This should enable, for example, neighbourhood policing and other agencies to make judgments with a better knowledge of the national security risk, and to implement appropriate local action."
Other areas likely to be covered in the strategy are efforts to improve the use of data by police and MI5, a new approach to managing the far-right threat, and increases to maximum sentences for some terror-related offences.
Mr Javid, who will attend a memorial service to mark the anniversary of the London Bridge attack on Sunday, said: "In May we mourned with Manchester. Today we remember the shocking attack on London Bridge and Borough Market.
"The Government is absolutely committed to doing everything possible to tackle the terrorist threat. It is my first priority every day in this job.
"We are working with the police, intelligence and security agencies, the private and public sector and international partners to make sure we have the best plans in place. I will be speaking about those plans in detail tomorrow when we publish our strengthened counter-terrorism strategy.
"But ultimately the strongest response is not just what we do, but who we are. The best way to stop terrorists achieving their aims is to stand by our values of tolerance, fairness and go out about our lives."