The BBC announced last week that the magazine show, hosted by Tim Lovejoy, Louise Redknapp and Simon Rimmer, will be cancelled after a run of six years.
The Facebook group 'SAVE Something For The Weekend From The Axe' now has over 22,000 followers.
Claire Lynch, who created the page, told the Radio Times "I would ask Mark Thompson to listen to the people who watch his channels and pay the licence fee.
"We understand that cuts need to be made and money needs to be saved, but taking away a programme that has constantly high ratings - and that people are willing to fight to save - isn't the way to do it."
Something for the Weekend's first programme of 2012, airing last Sunday between 10am and 11.30am, pulled in an overnight audience of 1m (12.2%), and was more popular than BBC One's Country Tracks between 11am and 11.30am.
Since news has spread of the axe, Professor Brian Cox wrote: "Very sad to hear that #STFW is going to be cancelled. I've enjoyed being on the show and watching."
Meanwhile, Lisa Faulkner said on the social networking site: "So so sad to hear that the brilliant something for the weekend is finishing. Top show. Boo."
Rumours emerged earlier today claiming that ITV may put in a bid to revive the programme.