Victorian Premier John Brumby says he is open to the idea of a drop box to allow people to legally abandon their unwanted babies.
It follows the discovery of a newborn boy's body in a shopping bag at a country bus stop near Shepparton on Tuesday.
Mr Brumby said there had been no proposals to the Government for a baby drop service, which had been set up in some countries, but he did not rule out the idea.
"I'd have an open mind on that issue," he said.
"It's a tragic case and obviously at the moment everybody is focused on finding the mother.
"These cases are always tragic, they're always very sad circumstances and we hope personally that we can find the mother, and I would say we've got an open mind on whatever steps or processes might be in place in the future to minimise or eliminate the number of occasions these circumstances occur."
In the US, a California hospital provides new mums with a how-to kit on abandoning a newborn safely and legally under the state's "safe surrender" laws, which aim to reduce infanticides and deaths from abandonment.
Last year, a Japanese hospital in the southern city of Kumamoto opened an anonymous drop box for unwanted infants, named "Stork's Cradle".
As well as protecting abandoned infants, it also aims to reduce the abortion rate.
Opposition Leader Ted Baillieu said a baby drop service would, in some ways, formalise what was already occurring.
"It's something that we would look at, we'd need to look at the detail of it, I know it's been raised many times before and there are plenty of agencies who've supported just that sort of thing," he said.
"In some ways that works now but it's not a recognised process and inevitably if a baby's abandoned ... it's a sad case but the baby will end up being supported by an agency."