A James Cook statue in Fitzroy Gardners on the edge of Melbourne's CBD has become the latest colonial monument targeted in a string of desecrations in just over a month.

At around 3:45am Monday morning, a group of four people allegedly cut down the statue at Cook's Cottage - a former residence of Cook's family transported to Naarm - now operating as a visitor's centre.

The site in East Melbourne has repeatedly been vandalised around 'Australia Day' on previous occasions.

CCTV captured the recent incident, which has been handed to police.

City of Melbourne Council are assessing the damage and will consider whether to repair the statue.

Police are investigating the matter.

"Officers were told unknown offenders attended the Fitzroy Gardens in Wellington Parade between 5pm on (Sunday) and 7am on (Monday)," a police spokesperson said, via the Herald Sun.

"It is understood the statue was damaged, causing it to fall to the ground."

According to reports, an anonymous post was made to social media from those responsible.

"Yet another monument to the imperialist James Cook has been felled in so-called Melbourne. Rumour has it that this was the last remaining Cook statue in the city," the post reportedly read.

"Monuments such as this only serve to prop up the narrative that enables so-called Australia's continuing theft and desecration of land and life, and to legitimise its ongoing violence.

"The words 'the colony will fall' were painted beside the fallen statue – because we know the colony can, must, and will fall. The legacy of the imposter Australian empire is slowly but surely crumbling around us, piece by piece – brick by brick – statue by statue."

Separate monuments of Cook and Queen Elizabeth were targeted between January 25-28.

The Cook monument at Victoria Gardens, in Melbourne's inner-North remains bare.