Plantasia is set to be shut as part of Â£81m in savings needed by the city council over the next three years.
The site at Parc Tawe attracts about 80,000 visitors a year, and its greenhouses are home to several endangered species and tropical plants.
A 'Friends of Plantasia' group says it wants to explore funding options to save the site.
The hothouses are home to cotton top tamarin monkeys, parakeets, bearded dragons, chameleons and a Burmese python, in addition to its collection of tropical plants and trees - including passion fruit, banana, fig and sugar cane.
"I feel very, very sad it's facing closure, Plantasia is such an iconic attraction," said Kate Cross, from the campaign group.
"The staff have always gone out of their way to be fantastically knowledgeable, very friendly and very approachable.
"We would like the City and people who have the expertise to get behind us and to offer us grants or funding - anything to preserve this attraction."
The group will meet next week to see if there is an option to establish a trust, and seek expert advice on funding.
Staff at the centre have already been served with 90-day redeployment notices, after Swansea council voted through the closure plans in February.
A spokesman said: "We have a budget deficit of Â£81m over the next three years and have to make difficult decisions to protect as many vital services as possible."
They said discussions with a number of interested parties and the private owners of Parc Tawe are on-going about the attraction's future.
"If Plantasia is closed then the continued welfare of its animals and plants will be of paramount importance," he added.
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