Instead, the group of television stars, politicians and campaigners will be muting, blocking and reporting "abhorrent" and derogatory comments - with the worst handed to the police - in a bid to starve so-called trolls of the wider audience they reportedly crave.
The move wants to stamp out entirely those who are using social media to spread racist, sexist, xenophobic and other hateful messaging via retweets and public shaming by well-known figures on social media.
Lineker, with his 7.4 million Twitter followers, is encouraging fellow social media users: "Don't rise to the bait, block the trolls and take some time out."
London Mayor Sadiq Khan, former home secretary Alan Johnson and ex-minister for business Margot James MP are among the politicians to have backed the initiative, while Dragon's Den star Deborah Meaden, Pointless quiz show host Richard Osman, former The Apprentice sidekick Nick Hewer and comedian Aisling Bea have also vowed to no longer engage with trolls.
The public figures have been convinced by new research that suggests hate speech is being inadvertently spread via social media when insults, put downs or worse are quoted or shared.
To reverse the worrying trend, the likes of Lineker and Riley have signed-up to instead reporting the worst cases of online abuse and vile messages to the police, while sending lesser examples to social media companies such as Twitter and Facebook to put pressure on them to act.
And new charity, the Centre for Countering Digital Hate (CCDH), the group behind the Don't Feed the Trolls report, is advocating an even simpler solution for those being badgered online - it recommends muting notifications and taking a break from social media in the first instance, before escalating anything thought to be unlawful.
In one eye-opening piece of analysis, a quoted tweet by Labour and anti-Brexit MP David Lammy to his 562,000 followers was able to increase his abuser's own popularity by 14 per cent. The abuser had accused the politician of not being "indigenous English".
Lineker, presenter of BBC's Match Of The Day and ex-Tottenham Hotspur goal scorer, said he was determined to "show online trolls the red card" after seeing the racist abuse directed at young black Premier League footballers.
Chelsea striker Tammy Abraham, 21, admitted his mother was in tears after reading the torrent of racist comments targeted towards him after he missed the decisive penalty in the European Super Cup against Liverpool last month.
Fellow England international and Manchester United marksman Marcus Rashford suffered similar backlash after missing a spot-kick against Crystal Palace in a league match this season.
Lineker said: "We've all been shocked by the way in which racist trolls have been targeting footballers recently.
"It is frankly horrifying that they have done so in a calculated way to spread their abhorrent views. Let's not allow the beautiful game to be tarnished in this way.
"Everyone across the sporting world will be grateful for this guide on how we can show online trolls the red card. Don't rise to the bait, block the trolls and take some time out."
CCDH's report - co-authored by charity chief executive Imran Ahmed and Dr Linda Papadopoulos, a psychologist featured on television and previously contracted by the Home Office - discovered neo-Nazi groups in America had actively encouraged supporters to target public figures in a bid to widen their exposure.
The report, published on Monday, contains excerpts from a playbook produced by the US neo-Nazi website, The Daily Stormer, advising that the best way to gain "media attention and general infamy" is "to troll public figures and get them to whine about it".
Jewish TV-personality Riley, a former Strictly Come Dancing contestant, has been on the receiving of anti-Semitic abuse and was involved in the research behind the report.
She said the experience had "totally changed the way I interact on Twitter".
"Before having CCDH's knowledge it felt like not responding to trolls or blocking them was weak, and calling them out, trying to engage in conversation and education was helpful, but the research shows otherwise," said the Celebrity Gogglebox star who has campaigned against anti-semitism in the Labour Party.
"I now block trolls as common practice, and have changed my settings to avoid seeing much of their output, which has made life much better from a mental health standpoint and vitally, is not inadvertently helping to grow their audiences or feed their negativity."
Mr Khan, Labour mayor of London, encouraged social media users to play their part in tackling online hate.
The former minister with Pakistani heritage said: "I've seen first-hand the online hate when social media is hijacked by hateful and cynical users who usually hide behind anonymous accounts.
"All of us who use social media have tremendous power in what we give our attention to and how we react to social media conversations.
"By ignoring, muting or blocking the trolls we can deny them the reactions they seek, while government and social media companies must up their game to ensure it is a safe space for people to exchange ideas."
Bea, an Irish comedian and writer of Channel 4 show This Way Up, previously campaigned in favour legalising abortion in her home country.
"We need to develop and practice our empathy, not just in real life but in our online lives which have escalated without rules of engagement, benefiting no one," she said.
Osman - host of Twitter's world cup of crisp flavours - said: "I love using social media, and I believe it can be a tremendous force for good. The key is to notice the intelligence and kindness, and to ignore the deliberate trolls.
"Mute them, block them, whatever you need to do. They are hungry for your attention and for your reaction. Don't feed them."