Wirrimbirra Sanctuary's animals are in the process of finding new forever homes.
The operators of the Bargo sanctuary, the David G. Stead Memorial Wildlife Research Foundation of Australia, were forced to relocate their native friends after receiving an eviction notice last month.
The Stead Foundation has been told to vacate the property and remove all animals by the start of October.
Foundation president David Stead it was a "distressing" situation.
"The animals' care is our primary concern," he said.
"The flight birds have all gone. Most wallabies are gone.
"The remaining wallabies and kangaroo will be gone by the end of next week.
"The emus and wallabies are our greatest concern."
Mr Stead said they were working with similar sanctuaries and other organisations to find suitable housing for the animals.
"Moving these very old and frail animals is a concern," he said.
"Warren the wombat is possibly the oldest living wombat in Australia, at nearly 38 years old.
"If necessary we will be asking for an extension, to ensure the animals are all clear of the property."
The sanctuary, along Remembrance Drive, has existed since 1962 as an education and conservation facility.
Thousands of school children have visited the site on excursions during the past five decades.
The National Trust (NSW), the owner of the site, has found a new operator for the sanctuary.
The Trust is expected to announce the preferred candidate for the lease of the property in late October.
The Stead Foundation's relationship with the National Trust has been strained for many years.
A National Trust (NSW) spokeswoman said the organisation would care for any animals at the site.
"We understand that at least some of the animals have been rehomed," she said.
"As stated previously, the National Trust (NSW) will ensure that if there are any animals that are unable to be removed from the property they will be appropriately cared for."
The National Trust spokeswoman confirmed the Bargo Dingo Sanctuary would remain at the property.
However, Mr Stead said the Trust had not provided the foundation with any information about how the other animals would be cared for if they were to remain at the site.
He said although animals were now leaving, the fight was not over.
"We will continue to fight our removal from the heritage property we created and its loss of heritage value, now and long after we are forced off the land," he said.
"This will continue through legal and political means until all avenues are exhausted."
A gofundme page has been set up by the Oolandra Wildlife Sanctuary to help remove animals from the site.
Oolandra operator Helen Stevens said the funds would be used to transport the animals.
"It can be an expensive process because some of the animals will need to be sedated," she said.
"It is also a 300-kilometre round trip for us to get them and as we are a volunteer-based organisation we need all the help we can get."
A public meeting attended by 70 people was held last month to protest the Wirrimbirra closure.
"There was good support, interest in the future of the animals and a general community concern for what will happen in the future," Mr Stead said.
"We have also met with the Wollondilly MP Nathaniel Smith.
"The best thing the local community can do if they want to support us is to contact Nathaniel's office."