The man arrested by police following the killing of the MP Sir David Amess has been named as Ali Harbi Ali.
The 25-year-old is being held under the Terrorism Act and officers have until Friday to question him.
The BBC understands Mr Ali was referred to the counter-terrorist Prevent scheme some years ago, but was never a formal subject of interest to MI5.
Whitehall officials told the BBC that the man being held was Ali Harbi Ali, a British man of Somali heritage.
Police said a man, who was held on suspicion of the MP's murder in Essex on Friday, was now being held at London police station after being rearrested under the Terrorism Act.
Early investigations revealed a potential motivation linked to Islamic extremism, police said on Friday.
Officials do not believe anyone else was involved in the attack.
Sir David, who had been a Conservative MP since 1983 and represented the Southend West constituency, was stabbed multiple times as he held a regular Friday meeting with constituents at Belfairs Methodist Church in Leigh-on-Sea.
It is thought Mr Ali did not spend long in the Prevent programme - which aims to stop people becoming radicalised.
Teachers, members of the public, the NHS and others can refer individuals to a local panel of police, social workers and other experts who decide whether and how to intervene in their lives.
Engagement in the scheme is voluntary and it is not a criminal sanction.
Mr Ali was initially arrested on suspicion of murder and held in Essex, but has since been transferred to London where he was further detained under Section 41 of the Terrorism Act.
On Saturday, detectives were granted a warrant by magistrates to allow them to keep Mr Ali in custody until Friday 22 October.
In a statement, the Metropolitan Police said a knife used in the attack had been recovered at the scene.
Officers spent Saturday searching three addresses in the London area. A post-mortem examination also took place on Saturday, the police added.
Sir David, 69, who was married with four daughters and a son, is the second MP to be killed in recent years following the murder of Labour MP Jo Cox in June 2016.
The latest attack has raised concerns for the safety of MPs, many of whom hold constituency surgeries which anyone can attend.
Home Secretary Priti Patel said MPs had access to a "panoply" of security measures - many of which were put in place after Ms Cox's murder - but said changes could be made to constituency surgeries.
However, Ms Patel warned that measures needed to be proportionate, telling the BBC's Andrew Marr show: "We're here to serve, we're here to be accessible to the British public."
During the interview she also recalled the moment she was told Sir David had been killed while she was at a cabinet meeting on Friday afternoon. "Our world was shattered," she said.
Meanwhile, Tory MP Andrew Rosindell said the killing of his friend and fellow Essex MP "shouldn't change things in a way that stops us going about our democratic role".
"There's got to be some balance to this. I don't have an answer," he told BBC Breakfast on Sunday. "This is not the Britain I want, this is not the country that we're used to."
Labour's Diane Abbott MP said she would prefer to meet constituents behind a screen to prevent possible stabbing attacks.
And Kim Leadbeater, the sister of Mrs Cox, said her partner had asked her to stand down as MP for Batley and Spen after Sir David's death.
Commons Speaker Sir Lindsay Hoyle said he was working "at pace" with the home secretary and police to find ways to improve safety for MPs.
Tributes to Sir David have been pouring in from politicians and constituents, with the home secretary calling Sir David a "man of the people" who was killed doing "a job he loved".
Ms Patel said he "was absolutely there for everyone, he was a much loved parliamentarian, to me he was a dear and loyal friend, but also he was a devoted husband and father".
A candlelit vigil was held in Leigh-on Sea on Saturday night to mark Sir David's life.
Speaking to the gathering, Southend councillor Alan Dear said the MP was "a gentleman and a gentle man".
BDST: 1700 HRS, OCT 17, 2021