‘Save Shawwood Greenspace’ was started by resident Kirsti McGinlay in response to the claims that Shawwood could be earmarked as a potential site for the new Eastwood Leisure Centre.
Since it began on Friday, it has gathered more than 750 members.
A similar page has been set up to oppose building on Broomburn Park.
Councillors are expected to consider seven sites for the potential new building in the coming months.
Those sites include Broomburn Park, Woodfarm Pavilion and playing fields, Overlee Pavilion and playing fields, Braidbar, Rouken Glen Park, Shawwood Park in Newton Mearns and Eastwood Park.
Kirsti said: “I was stopped by a neighbour who said that Shawwood looked likely to be the location of the new built.
“The consultation was very misleading. What we are planning to do next is chap on doors to get as much support for this as possible. We heard from a lady on Facebook who went through something similiar, so we have all the residents’ support.”
Shawwood Park is frequently used by dog walkers and children.
Kirsti continued: “I have children and a dog, and we use the park to let the dog just run and run. Other dog walkers use it as well as local children who use it to play.
“You might not see hundreds of children at one time, but it is frequently used.”
Another issue raised by Kirsti is the issue of increased traffic in the area.
She said: “Traffic flow is going to be horrific. Would they use the entire green belt and open up a through road all the way through to Eaglesham Road? We don’t know because the plans are not in the public domain.
“Shawwood Park is a jewel in our community that should be enjoyed by all our residents.”
For further information, visit www.facebook.com and search ‘Save Shawwood Greenspace’ and ‘Broomburn park’.