Sir David, 69, who had been an MP since 1983, was fatally injured while meeting constituents.
A 25-year-old man arrested at the scene on suspicion of murder is in custody at an Essex police station.
The name the police have for the suspect is Ali Harbi Ali, according to the Press Association.
Official sources say the man is believed to be a British national with Somali heritage.
The investigation is assuming there was an Islamist extremist motivation for the attack, official sources said.
But it is understood the suspect was not, and had not previously been, a subject of interest for the security services.
According to reports, some sources have claimed the suspect has the same details as a man previously referred to Prevent, the anti-terror programme for those thought at risk of radicalisation. But there is yet to be any official confirmation of whether this is the case.
As part of the investigation, officers are also carrying out searches at two addresses in the London area, the Met Police said.
new video loaded: Britons Pay Tribute to Slain LawmakertranscriptBackbars0:00/0:53-0:53transcriptBritons Pay Tribute to Slain LawmakerBritish leaders and local residents paid their respects to David Amess, a Conservative lawmaker who was fatally stabbed on Friday while meeting with constituents at a church in Leigh-on-Sea, England.I feel that I have lost a family member. I feel that he was a family of Southend, he was the leader of Southend, and he was everywhere. […]