Vandals who struck in the dark have tried to decapitate a statue of Captain Cook and sprawled graffiti on the historical cottage his father built 270 years ago.
Victorian Premier Jacinta Allan hit out at the "senseless vandalism", saying it "has no place".
"It advances no cause," the premier told reporters in Brunswick on Tuesday.
"We'll work with council authorities around any restoration of repair work that needs to be undertaken."
Victoria Police say two offenders arrived under the cover of darkness at Fitzroy Gardens at about 2am and spray-painted "anti-Cook comments" on the cottage.
"The Captain Cook statue, which also stands in the gardens, has been damaged by what appears to be an attempt to remove the head," a police spokesman said.
The vandals left suddenly after security staff notified them they were being recorded, Melbourne Lord Mayor Nick Reece said.
“While there are a range of views on statues and memorials, each time a monument is damaged, it’s ultimately the ratepayer footing the bill – and that is unacceptable," he said.
The cottage was built by James Cook's father in 1755 in Yorkshire, England, and was transported to Melbourne by Sir Russell Grimwade in 1934.
The incident comes just a month after another statue of Captain Cook was splashed with red paint and had its hand and nose knocked just before the Australia Day weekend.
Last year a statue of Captain Cook was hacked off at the ankles in St Kilda by activists on the eve of Australia Day.
A few months later In June, a monument of King George V was beheaded in the early hours of the King's Birthday public holiday.
Investigators are checking CCTV footage and have appealed to any witnesses or those with information or dashcam footage to call Crimestoppers.
