The investigation into the fatal stabbing of Sir David Amess is being led by counter-terror officers, Essex Police have said.
The death has led to Home Secretary Priti Patel asking all police forces to review security arrangements for MPs "with immediate effect."
Chief constable Ben-Julian Harrington said 69-year-old Southend West MP Sir David was "simply dispensing his duties when his life was horrifically cut short."
A 25-year-old man, believed to be a Briton with Somali heritage, has been arrested on suspicion of murder following the attack at Belfairs Methodist Church in Leigh-on-Sea near Southend at midday on Friday and remains in custody.
Police said the early investigation has revealed “a potential motivation linked to Islamist extremism.”
Harrington said: "The investigation is in its very early stages and is being led by officers from the specialist counter-terrorism command.
"We made it clear at the time of the incident that we did not believe there was any immediate further threat to anyone else in the area.
"It will be for investigators to determine whether or not this is a terrorist incident, but as always they will keep an open mind."
Tory veteran Sir David, who was described by Prime Minister Boris Johnson as "one of the kindest, nicest, most gentle people in politics," had been an MP since 1983 and was married with five children.
Harrington said officers were called to reports of a stabbing at just after midday.
He said: "The response of the emergency services to this incident was immediate and our officers arrived on scene within minutes.
"When they arrived they found Sir David Amess MP, who had suffered multiple injuries.
"This was a difficult incident, but our officers and paramedics from the East of England Ambulance Service worked extremely hard to save Sir David.
"Tragically he died at the scene. A 25-year-old man was arrested immediately at the scene on suspicion of murder. He remains in custody.
"A knife was also recovered at the scene."
A spokesman for Home Secretary Patel said she chaired a meeting of the Police, Security and Intelligence Agencies to discuss the incident in Southend and the ongoing response, adding that she had also spoken to the Speaker of the House of Commons.
"The Home Secretary has asked all police forces to review security arrangements for MPs with immediate effect and will provide updates in due course," the spokesman said.
Tributes have been paid to Sir David since his death was confirmed, with Prime Minister Johnson saying: "I think all our hearts are full of shock and sadness today at the loss of Sir David Amess MP, who was killed in his constituency surgery in a church after almost 40 years of continuous service to the people of Essex and the whole of the United Kingdom.
"And the reason I think people are so shocked and saddened is above all he was one of the kindest, nicest, most gentle people in politics, and he also had an outstanding record of passing laws to help the most vulnerable, whether the people who are suffering from endometriosis, passing laws to end cruelty to animals, or doing a huge amount to reduce the fuel poverty suffered by people up and down the country.
"David was a man who believed passionately in this country and in its future.
"And we've lost today a fine public servant and a much-loved friend and colleague, and our thoughts are very much today with his wife, his children, and his family."
A vigil, led by Father Jeffrey Woolnough, is being held at St Peter's Church on Eastwood Lane, in Leigh-on-Sea.
A photograph of Sir David has been placed at the front of the church.