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.

More than 300,000 children under the age of five are severely malnourished and require urgent assistance.
Most of those in need of help have been internally displaced following decades of conflict.
Malnutrition levels in Somalia have increased over the last six months with nearly half the population affected.
The number of people without enough food has increased by 300,000 since February.
Somalia drought leaves children 'facing death'
Peter de Clercq, the UN humanitarian coordinator for Somalia, said those providing aid were ready to "scale up" their response to help families struggling to find food.
But he noted that funding for the Somalia Humanitarian Response Plan has reached just 32% of its target.
The crisis follows poor rainfall in the south and central parts of Somalia where crop production has fallen by more than half in just six months. Farmers have also lost many of their livestock in the recent drought.
Aid agencies have warned that the numbers could go higher if the Dadaab refugee camp in Kenya is closed down and thousands are forced to return to Somalia where they have no homes or livelihoods.
In January, aid agencies launched an appeal for more than $880m (Â£680m) to alleviate the humanitarian situation in Somalia but, so far, less than half of that sum has been donated.
The UN report, released on Tuesday, suggests that crop production was severely hit by insufficient rainfall and flash flooding in southern and central Somalia, "the breadbasket of the country".
The agency said continued population displacement following decades of conflict and attacks by extremist groups such as al-Shabab was a major contributing factor.
Who are Somalia's al-Shabab?
Al-Shabab, the Islamist militant group, has been battling the UN-backed government in Somalia for years and has carried out a string of attacks in neighbouring Kenya.
The group, which is allied to al-Qaeda, has been pushed out of most of the main towns it once controlled, but it remains a potent threat.

On Sunday, BBC 6 Music DJ Mark Radcliffe explores the history of the football chant to find out what makes a hit in the stands.
Before the programme goes live, we want to test your chant knowledge. Listen to these seven terrace chants and work out which club's fans are singing them.
You can listen to 'From Doris to Depeche: The football fans' songbook' on BBC Radio 5 live from 19:30 BST.
An indoor 'rainforest' under threat of closure in Swansea could be run as a trust, say campaigners.
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Nearly five million people in Somalia are suffering from a shortage of food due to poor rainfall, floods and displacement, the United Nations says.
[NEXT_CONCEPT]
From Blue Moon to You'll Never Walk Alone, almost every team have their own song - but do you recognise some of the lesser-known chants?