The posting, which provided a picture of the infant, was brought to police attention on Monday.
Victoria Police Const. Michael Russell says officers are trying to track down the parent and confirm that no child is at risk.
Police didn't provide other details of the posting and the ad was quickly removed from the Used Victoria website, where new and used goods are sold.
However, an anonymous person later replied to the parent who wanted to give away baby Alex — according to the original post — "because times are hard."
The reply urged the parent to reach out for help or to drop the baby off in a safe location if they truly couldn't look out for the child.
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