Nine terror plots have been foiled in the UK over the past year, cabinet ministers have been told.
Ministers were told that the IS forces had been beaten in Iraq in Syria but were now pushing for attacks to be carried out on UK soil.
It comes as Home Secretary Amber Rudd prepared to give a statement to Parliament on the findings of official reviews into counter-terror agencies handling of this year's attacks.
Mr Parker told the Prime Minister that five attacks have been successful over the same period, four of which related to Islamist terrorism.
Ms May's spokesman said: "The Prime Minister led thanks to the tireless work of staff at MI5 to combat the unprecedented terrorist threat.
"Cabinet ministers heard that while Daesh suffered major defeats in Iraq and Syria, this did not mean the threat is over. Rather it is spreading to new areas, including trying to encourage attacks in the UK and elsewhere via propaganda on social media."
Home Secretary Ms Rudd said during the briefing on Tuesday, that the pace of "attack planning" had increased significantly over the year and that the Government had been pressuring social media companies to remove extremist material.
She was set to speak to MPs in a Commons session shortly after as the findings of official reviews into intelligence handling this year's wave of attacks.
Britain's security apparatus came under close scrutiny after dozens of people were killed or injured in atrocities between March and June, including those at Westminster, Manchester, London Bridge, and Finsbury Park.
MI5 and police launched internal reviews to consider what was known about the perpetrators prior to the attacks, examine decisions made regarding any intelligence held before the attacks took place and identify any areas for improvement, such as in the management of thousands of individuals who have been looked at as part of previous inquiries but who are not currently under active investigation.
The assessments have been carried out internally but David Anderson QC was appointed by the Government to provide "independent assurance".
Mr Anderson, the former independent reviewer of terrorism legislation, submitted his classified report to the Home Secretary last month and a version of his findings will be made public on Tuesday.
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