NekNominationinvolves participants filming themselves downing deadly amounts of alcohol, posting the videos online and nominating friends to do the same.
The Local Government Association (LGA) say that the likes of Facebook and Twitter should warn their users against participating in the craze.
Katie Hall, a member of the LGA, said: “This is an utterly reckless and totally irresponsible craze which has tragically claimed lives. More should be done to highlight the dangers and persuade people not participate.
“We believe social media operators have the responsibility to provide health warnings to user groups and individuals.”
And Katie said that the LGA wants to work with the social networking sites to tackle the problem together.
She said: “The LGA is looking for these corporations to show leadership – and not ignore what is happening on their sites.
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“We are urging Facebook and Twitter executives to sit down with us and discuss a way forward which tackles this issue head-on.”
It was shockingly revealed this week that the deadly craze has spread to primary schools, with a boy just 10-years-old posting a video of himself downing a cocktail of sauces, cream and vodka.
The boy, despite being fine in the end, was violently ill and had to see a doctor.
Following the five deaths, users are now being warned they could face legal action if they nominate people to take part in the game.