Boris Johnson will lead tributes to Sir David Amess in the House of Commons on Monday as debate rages about how drastically to step up security in the wake of the fatal attack on the Southend MP at his constituency surgery.

On Sunday night Amess’s family appealed for public unity, urging people to “set aside their differences and show kindness and love to all”. In a statement, his relatives said they were “absolutely broken” but had drawn strength from the tributes to him from across the political spectrum.

The attack sent shock waves through Westminster and reopened questions about MPs’ safety five years after the murder of Labour’s Jo Cox. The home secretary, Priti Patel, said on Sunday that she was considering offering MPs police protection at their surgeries, and the use of airport-style scanners was under consideration.

Asked how quickly such measures could be brought in, Patel told Sky News that all MPs were being contacted by their local police forces. “This isn’t a case of let’s wait for two weeks, three weeks, four weeks. These are immediate changes, and measures that are actively being put in place, and it starts with MPs.”

However, several MPs told the Guardian they had concerns that a police presence would deter constituents from attending surgeries or other public events. The former Brexit secretary David Davis said: “The people who come to your surgery are people who are at their wits’ end: they’ve been let down by their employer or their doctor or the NHS or the welfare system, and they’re often very fragile. They might well be put off by a big burly police officer on the door.”

The former shadow home secretary Diane Abbott, who represents Hackney North, said MPs might benefit from access to metal detectors or wands to check constituents, or plastic screens, but “if you put police officers outside our constituency advice surgeries it makes us look like agents of the state – and in Hackney that is not a good look.”

A senior backbench Conservative pointed out that resources would have to be directed at the MPs most under threat, but Amess would have been unlikely to have been identified as an obvious target.

Others said security measures put in place in the wake of Cox’s murder had only been partially implemented, in some cases because of a lack of police resources. Several told the Guardian they did not have the single point of contact at their local force that is meant to be at the heart of the system.

The shadow home secretary, Nick Thomas-Symonds, said: “It is right that we look at a range of options, to ensure we get the balance right between keeping MPs safe whilst doing their jobs, giving them confidence in the support available and to protect the unique nature of British democracy.”

A spokesperson for the parliamentary authorities, which oversee MPs’ security in cooperation with the police, said: “It is essential that we learn from this tragic event, identify any additional security requirements, and continue to encourage MPs to take up the existing measures available to them.”

A 25-year-old man, Ali Harbi Ali, a British national, was still being questioned at a London police station on Sunday in an investigation led by counter-terrorism officers from the Met. He was arrested on suspicion of murder on Friday after being detained by officers at the crime scene in Leigh-on-Sea, Essex.

Police can keep Ali in custody until Friday before deciding whether to charge him, under powers granted by terrorism legislation. The killing was assessed as being linked to an Islamist ideology because of developments in the investigation after the arrest, sources say.

A home in Kentish Town, north London, where Ali and his family are believed to have lived, was still being searched by police on Sunday. Two other addresses in the London area have been searched by counter-terrorism police.

The suspect was previously known to the Prevent scheme, the government programme to stop radicalisation, but his involvement was short, according to multiple sources. He did not appear on any current MI5 watchlist, sources added.

Ali’s father, Harbi Ali Kullane, is a former adviser to the prime minister of Somalia and now lives in the UK. He told reporters on Saturday he was feeling “very traumatised” by the violent incident.

Sources close to the investigation indicated on Sunday that Ali had booked an appointment to see Amess at his surgery on Friday. Details of the surgery had been advertised on social media and elsewhere in advance.

The Speaker of the House of Commons, Sir Lindsay Hoyle, has torn up the parliamentary timetable to allow MPs to spend much of Monday afternoon paying tribute to Amess, after a minute’s silence at 2.30pm. Johnson will lead a special debate in which MPs will be able to share their memories of the Essex MP, who was first elected in 1983.

Several including Hoyle have already made clear they believe the best tribute to him would be to carry out his long-held wish that Southend become a city, a hope that was echoed by Amess’s family in their statement on Sunday. “David was working hard for Southend to gain city status. In his memory, please show your support for this campaign,” they said.

As well as strengthening physical security around MPs, Patel suggested the government was looking at ways to ensure social media companies do their part in tackling what she called the “corrosive” state of online debate.

“We can’t just apply a binary approach, but there is something very, very corrosive,” she said. “We know that social media platforms advocate all sorts of things that are harmful to all aspects of society,” Patel said, adding that it was important to “really close that corrosive space where we see just dreadful behaviour.”

The government’s online safety bill is being scrutinised by MPs and there have been calls for it to be toughened up. Labour would like to see social media executives made personally responsible if their companies fail to uphold the codes of practice in the legislation.

Amess’s family paid tribute to his strength and courage in their statement. “He was a patriot and a man of peace. So we ask people to set aside their differences and show kindness and love to all. This is the only way forward. Set aside hatred and work towards togetherness. Whatever one’s race, religious or political beliefs, be tolerant and try to understand.”

They added: “As a family, we are trying to understand why this awful thing has occurred. Nobody should die in that way. Nobody Please let some good come from this tragedy. We are absolutely broken, but we will survive and carry on for the sake of a wonderful and inspiring man.”

On Sunday, the state of Qatar issued a statement condemning the attack on Amess, who was chairman of the All-Party Parliamentary British-Qatar Group, calling his stabbing “a horrific crime and a clear violation of human rights”. Amess had visited the nation last week.