A controversial monument from colonial-era Victoria has been cut in half and an ANZAC memorial has been doused in paint in the latest in a string of vandal attacks on statues.

A memorial honouring one of Melbourne's founders, John Batman, at Queen Victoria Market was broken in the early hours of Saturday - the first day of the Australia Day long weekend.

"Police are investigating after a statue was damaged in Melbourne's CBD on 25 January," Victoria Police said in a statement.

"Officers were called to gardens on Queen Street following reports a statue of John Batman memorial statue had been damaged about 2.20am."

Much of the damaged plinth memorial near the car park had been removed by Saturday afternoon, with a loose cloth covering its base.

Mr Batman was a leading figure in the establishment of Victoria in the 19th century.

According to the National Museum of Australia, John Glover, a neighbour of Mr Batman's while he was living in Tasmania, once described the colonial figure as 'a rogue, thief, cheat and liar, a murderer of blacks and the vilest man I have ever known'.

Unveiled in 1881, the monument to him originally said the land was "then unoccupied", according to Melbourne City Council.

A plaque was added in 1992 specifying Aboriginal people had originally occupied the land, with a second installed in 2004 to acknowledge the statue's inaccuracies.

Police are also probing the theft of two ceremonial flags and vandalism to an outdoor stage in Ringwood in the city's east about 2am on Saturday.

Images posted to social media showed vandals had spray painted the words "the colony will fall" and "this land is stolen".

A probe is also under way after a memorial to ANZAC soldiers in Parkville, in inner-city Melbourne, was covered in red paint about 3am on Saturday.

Red spots and a smattering of colour remained at the solder's feet on Saturday afternoon after most of the paint was washed off.

Victoria Police has not linked that vandalism attack to other incidents.

City of Melbourne Lord Mayor Nick Reece said security around "high-risk" targets had been temporarily ramped up.

"Defacing and damaging city assets will not be tolerated in Melbourne," he said.

"Our team has responded swiftly and professionally to these attacks, and cleaning and repairs were under way within an hour of the city being notified."

It comes after four people decapitated the statue heads of two Australian prime ministers at the Ballarat Botanic Gardens, causing more than $140,000 in damage.

The heads of former Labor prime ministers Paul Keating and Kevin Rudd were removed and stolen on Thursday morning, while the name plates of the 18 remaining statues were covered in spray paint.

In Sydney, a sandstone heritage statue of British explorer and naval captain James Cook in Randwick had its hand and nose severed and was partially splattered with red paint on Friday morning.

Anyone with information about any of the incidents has been urged to contact investigators.

Rachael Ward - AAP