The Home Secretary said on Sunday a “whole spectrum” of measures are being considered to address safety concerns in the wake of the Southend West MP’s death on Friday.
It came as Labour shadow foreign secretary Lisa Nandy admitted she did not feel safe when going about her Wigan constituency.
Ms Patel said: “We need to close any gaps, basically, where we feel that there are concerns.
“This isn’t a case of let’s wait for two weeks, three weeks, four weeks; these are immediate protective measures.
“There are ways in which we can do things differently, clearly around surgeries.
“You’ve just mentioned publicising them, moving from publicising appointments to pre-booking appointments, making sure that appointments are checked thoroughly, that backgrounds on individuals are checked.”
MPs could also be asked to share their whereabouts at all times with the police in a bid to keep them free from harm, she said.
Asked if she would consider airport-style security, Ms Patel said: “That would be with the police and the House authorities. There are lots of things under consideration already.”
Ms Patel also explained the security forces have been watching for people who may have become radicalised online during the coronavirus lockdowns and could pose the threat of a ‘lone-wolf’ attack.
The Home Secretary said: “It is a really important point – the coronavirus pandemic, people being locked down at home, online etcetera.
“But to put this into perspective, we have the best security and intelligence agencies in the world.
“I know – I can’t sit and share with you – how they have worked throughout the pandemic, I know the work they do in watching individuals, subjects of interest, tracking behaviours, monitoring anybody of interest.
“Threats are always there and if you listen to my colleagues, even the director general of MI5, he has spoken publicly about lone actors.
“We’ve seen a lot more of that, at both ends of the spectrum by the way – Islamist and extreme right.”
Conservative Sir David, 69, who had been an MP since 1983, was meeting constituents at Belfairs Methodist Church in Leigh-on-Sea, Essex, on Friday afternoon when he was stabbed multiple times in a frenzied attack.
His death comes after the Labour MP for Batley and Spen, Jo Cox, was murdered in 2016 as she was on her way to a constituency surgery.
A 25-year-old man, named as Ali Harbi Ali, was arrested at the scene of the attack in Essex on suspicion of murder. He has since been detained under section 41 of the Terrorism Act 2000 and is in custody at a London police station.
A warrant of further detention, which allows detectives to hold the man until Friday, was granted at Westminster Magistrates’ Court on Saturday.
The shadow foreign secretary said that Sir David Amess’s murder was “heart breaking”.
Asked whether she felt safe doing her job in her constituency, the Labour MP for Wigan replied: “No, not really if I’m honest. I feel quite fortunate to have a lot constituents who are concerned about my safety.
"It does make you very anxious but none of us are going to stop doing our job as a consequence.
“Wigan is that kind of place, people look after each other and when they see things like this, my inbox this morning is absolutely full of people saying: ‘keep going and thank you’, and I’m sure lots of MPs are getting the same thing.
“But this isn’t the first time this has happened.”
Ms Nandy said the response to MP security concerns had been “far too patchy”, adding: “We can’t afford to get into a place where we are all trying to out do each other about the risks we are prepared to do this job – an attack on MPs is an attack on democracy because it silences the people who we represent, so we have to take this more seriously.”
The senior Opposition figure also admitted she was not sure the threat to MPs could ever be eliminated.
She explained: “We’ve had Stephen Timms, one of my colleagues who was stabbed a few years ago, we had the case of the Lib Dem MP before that whose case worker was killed, we’ve had a plot to kill Rosie Cooper, my neighbouring MP, and of course what happened to Jo and now David.
“I’m not sure that that moment is recoverable. I say that with sadness but I think there is an element of realism about that.
“MPs are well known in our constituencies, people tend to know where we live, we are out and about, we’re normal human beings, we go out and about on the weekends and go to the local shops.
“I’m not sure that we can ever eliminate the risk but there are other things that can be done to reduce the risk: I think the suggestion from the Speaker about ensuring that anyone who wants or needs security at surgeries is a good idea, not least because people often know, even if we don’t advertise them, that they are happening so they can become a magnet for people who want to come and cause trouble.”
It comes as MSPs received a letter from the presiding officer saying the Scottish Parliament would offer "advice and support" following the attack.Holyrood presiding officer Alison Johnstone said: "Whilst all our thoughts are with Sir David's family, friends and colleagues, it is understandable that at a time like this we reflect on our own work and the challenges we face.
"Representing our constituents is one of the greatest privileges of being a Member of the Scottish Parliament.
"But it is one that, sadly, can bring with it threats and fears for the safety of ourselves, our staff and families."