The move in by the Leichhardt local government area, in the city's inner west, came after deputy mayor Michele McKenzie's son rescued a brushtail possum and its baby when they were attacked by a cat.
There are also plans to reintroduce a colony of ringtail possums, which are much smaller in size, to the municipality, and Mr McKenzie wants the animals to have the best possible chance at survival.
But the odds are stacked against the possums unless cats, which can climb trees to catch the animals, are stopped from prowling at night.
There have been some 564 reported cat attacks on ringtail possums in the last year alone, making up the biggest part of total cat attacks on wildlife, according to wildlife protection agency WIRES.
"Its a small thing to ask people to keep their cats indoors if it means protecting our native wildlife." Mr McKenzie said