The man suspected of stabbing three people to death in a Reading park is believed to be Khairi Saadallah.
He is a 25-year-old Libyan refugee who was granted asylum in the UK, Sky News understands.
It is understood he was briefly on the radar of MI5 last year.
But there was not enough concern to launch a full investigation.
Police declared the stabbing incident in Forbury Gardens in central Reading at 7pm on Saturday a terror attack and counter-terrorism police have now taken over the investigation.
Saadallah was apprehended within five minutes of police being called at 6.