The 69-year-old, who is the MP for Southend West in Essex, was attacked at Belfairs Methodist Church, in Leigh-on-Sea, on Friday.
Essex Police confirmed a 25-year-old man has been arrested on suspicion of murder, and said a man had sadly died at the scene.
Sources have confirmed the man is Sir David, whose death comes five years after Jo Cox was murdered in June 2016.
Tributes have poured in from across the region to the Southend West MP who has been described as a "real gentleman" by colleagues.
Stuart Anderson, MP for Wolverhampton South West, said: "To be honest, I'm still trying to get my head around how anyone can do this. It's despicable.
"I'm devastated – words can't express, to be honest, the anger of why someone can do this. I feel for his family and the community.
"David was doing his job and doing his job well and it's cost him his life. As a soldier, you knew when you went overseas there was a chance you wouldn't come back – but he's an MP and this in the UK.
"I'm very concerned this happened with Jo Cox and it's still happening today. I'm still getting threats myself and it's a regular part of what we deal with as MPs. It's become a toxic part of politics."
Wolverhampton South East MP Pat McFadden, said: “It is hard to believe that another MP has been murdered just doing his job, helping his constituents at a local advice surgery.
"Sir David Amess had been in public service for forty years. He was one of Parliament’s longest serving MPs. MPs want to be available to the public, to help people with their problems.
"That’s why we hold advice surgeries like the one Sir David was holding when he was killed. I cannot imagine the pain his family is going through right now. May God rest his soul."
South Staffordshire MP Gavin Williamson said Sir David was a "kind, gentle and thoughtful man" who always put other people first and had helped him to settle in as a new MP.
"It is such a tragedy and my heart goes out to his family," he said. "As politicians we want to be there for the public and to be accessible. It is so awful when something like this happens."
West Midlands Mayor Andy Street said Sir David's death was a tragedy and described the attack on him as "truly abhorrent".
"We don't yet know the full circumstances, but all politicians will be asking if this is also an attack on the principle of free speech in the UK," he added.
Former Labour deputy leader and West Bromwich East MP Tom Watson said on Twitter: "This is overwhelming news. My love, thoughts and prayers are with David’s family, his parliamentary colleagues and many friends."
Michael Fabricant, MP for Lichfield, said: “The news that David Amess has been murdered while performing his duties as an MP is a shock to all decent people everywhere.
“He was a respected man beloved by all who knew him. May his soul rest in peace.”
Suzanne Webb, MP for Stourbridge, described it as a tragedy and offered her thoughts and prayers with Sir David's family.
She said: "This is a terrible tragedy and it has happened to such a kind man going about his job as a public servant. There are no words I can adequately use to explain how upset I am at this news."
Amanda Milling, MP for Cannock Chase, offered her thoughts and prayers with the MP's family, friends and staff.
She said: "Absolutely horrified to hear about the death of David Amess. Truly shocking this happened while he was doing his public duty as an MP and working to improve the lives of his constituents."
Lord Ian Austin, former MP for Dudley North, said: "Heartbreaking and horrifying news to hear he has died. Such a nice man.
"Always had a smile on his face. Loved life, loved his constituents and loved serving the public. Dreadful to think what his family must be going through."
Former Wolverhampton North East MP Emma Reynolds said: "Senseless violence. Horrific consequences. David Amess MP dedicated his life to serving his constituents for decades.
"He was a kind and generous man. Rest in peace David. You will be missed and deepest sympathy to your family."
James Morris, MP for Halesowen and Rowley Regis, said: "I am devastated to hear of the news of Sir David Amess, who was a very popular colleague to work with. My thoughts are with his family and those who knew him at this terrible time.
"Sir David was doing what he did best, representing the people of Southend West. It is appalling that he was attacked like this going about his business representing his constituents."
Nicola Richards, MP for West Bromwich East, said: “Deeply shocking and upsetting news about Sir David Amess MP.
“He was a fantastic constituency MP and such a warm friendly colleague to so many, even us newbies who hadn’t had the chance to get to know him as well as others.
“Thoughts are with his family, friends and staff.”
Eddie Hughes, MP for Walsall North, said: "We [have] lost a dear friend and colleague in the most dreadful of circumstances. David Amess was the kindest of men. He will be greatly missed.
"My thoughts and prayers are with his family at this terribly sad time."
Mike Wood, MP for Dudley South, offered condolences to the MP's family and said: "I really don't know what to say, it's shocking. This shouldn't be happening.
"I've worked with David over several years and I can't remember anyone, from both sides of the House, saying anything bad about him.
"No MP gets everything right but David was doing this job, doing this advice surgery and was trying to help his constituents. And for this to happen, it's hard to make any sense of it at all.
"I think political language has become less tolerant and more aggressive over the last few years or so and there's a big difference between criticising someone's politics and this violence.
"I think some of the way that political arguments are conducted, particularly on social media, does escalate and creates an environment where this is sadly more likely to happen but thankfully, it's extremely unlikely.
"I've been there six-and-a-half years, I was elected in 2015, and there's been two MPs killed in this time at their surgeries trying to help the people they represent."
Shaun Bailey, MP for West Bromwich West, said: "Obviously, it's absolutely horrifying and it's so sad because David was such a cherished colleague and a friend to everyone.
"The brief time I knew him, he was an utter gentleman with me. I remember him sitting there [as Mr Bailey told the MP of his background] and the interest he took, for someone who has been there for 40 years, was incredible.
"He will be so so missed and my thoughts are with his family."
Jane Stevenson, MP for Wolverhampton North East, said: "I'm shocked and horrified that Sir David Amess has died after being attacked today while holding a constituency surgery.
"My thoughts and prayers are with his family and all those who loved him.
"Since my election in 2019 I have worked with Sir David on the APPG for Fire Safety, which he chaired with great commitment and skill.
"We also worked together on a number of animal welfare issues. He was a caring, thoughtful and brilliant colleague, who will be hugely missed.
"Only last week at Party Conference in Manchester David and I spoke at an animal welfare event. He spoke, as ever, with clarity and passion for the cause.
"I will really miss his kindness and his sense of fun. I know the whole of the House thought well of him - an MP who cared deeply about his constituents, and added so much to our work in Parliament.
"RIP, dear David."
Theo Clarke, MP for Stafford, said: "I am really shocked at the murder today of my colleague Sir David Amess. He was a brilliant MP for Southend and was a pleasure to work with.
"He always made a huge effort to make us newbies feel at home in Parliament. Thinking of his family, friends and team on such a tragic day."