Melbourne's Yarra City Council will consider their next steps concerning a Captain James Cook monument after it was vandalised over the January 26 long weekend.

The stone monument at Edinburgh Garden's Rowe Street entrance in Fitzroy North was cut from its foundations, knocked over and spray painted with the words "COOK THE COLONY" on Sunday, two days after the public holiday.

It came amid a string of similar incidents to colonial memorials across the city - including another James Cook monument in St Kilda and a Queen Victoria statue close to the CBD.

Police are said to be investigating, with considerations to whether the three cases are linked.

The site in Fitzroy North has been subject to similar acts in recent years.

As late as Sunday evening, hours after it was targeted, the monument remained at the fork in the path at the busy park.

The monument has been removed for the time being, with no decisions made over its future.

"Last week, a monument in Edinburgh Gardens commemorating Captain Cook was vandalised," the spokesperson said.

"For the safety of the community, Council officers have since removed the monument.

"In the meantime, Council will assess the damage to determine next steps and keep the community informed."

On Monday, The Age reported council officers had recommended permanently removing the monument in alleged emails obtained by the outlet, with conversations already sparked ahead of the latest vandalism.

"Due to the seriousness of the new damage, officers strongly recommend that the memorial be removed from location and that the object be deaccessioned from the collection in line with policy processes," the email read, according to The Age.

"Deaccessioning is the formal process of removing objects from a collection."

Yarra councillor Stephen Jolly reportedly said he was not surprised the monument has been targeted with "increasing numbers of people, especially young people, go to the Invasion Day march and don't necessarily see it (January 26) as a day of celebration".

On Tuesday morning, Yarra Residents Collective spokesperson Adam Promnitz suggested removing the monument would be showing to "let the vandals and criminals win" in an interview with Melbourne's 3AW radio.

"It's criminal damage…If you're just gonna let vandals and criminals win, then you might as well just give up altogether," he said.

Mr Promnitz said the monument's targeting doesn't necessarily reflect the wider views of the community regarding January 26.

"There is a wide range of views on Australia Day, and that's not necessarily the debate happening here," he said.

"But I think most residents would turn around and go, 'You know what, this isn't the right way to have the discussion. This isn't the right way to do things'.

"You don't just get your own way by going and being destructive and antisocial and causing criminal damage."

He was later critical of the Council's spending on public facilities more broadly.

National Indigenous Times contacted Yarra City Council for further clarification regarding The Age's report. The Council did not comment further to their statement.