Source: Topple the Racists
Cook's arrival in New Zealand sparked a series of skirmishes that resulted in Maori deaths, which Britain signalled its regret over last year.
The Topple The Racists website says, "we believe these statues and other memorials to slave-owners and colonialists need to be removed so that Britain can finally face the truth about its past – and how it shapes our present".
There are statues of Cook across the UK and Conservative MPs have voiced fears that others will be targeted.
Dr Craig Barker, an archaeologist and a manager at the University of Sydney's Chau Chak Wing Museum, said Cook was mainly in the crosshairs for "the historical ramifications of his voyages".
Dr Barker said he did not think Cook statues had been singled out before in the UK because he is not as polarising a figure there as he is in Australia.
"To the average person in Britain, they have not been aware of the more nuanced conversations that we've had in Australia and New Zealand and the Pacific over the past decade about the rights and wrongs of his actions," he said.
"It shows that there is a ramping up of the debate about how these people are recognised by history and whether these memorials are an accurate or worthwhile way of recording these people."
Over the weekend, a statue of 17th-century slave trader Edward Colston was toppled in the UK city of Bristol, with statues deemed racist also being damaged around the US and Europe.
In Australia, some Indigenous groups are calling for "racist statues" to be taken down.
But on Thursday, Prime Minister Scott Morrison dismissed the idea that Cook statues should be removed, telling people to "get a grip".
Mr Morrison said Australian protesters raised fair issues about Indigenous prison rates and deaths in custody, but said the movement was being hijacked by radical left-wingers to push other causes.
"This is not a licence for people to just go nuts on this stuff," Mr Morrison told 2GB.
He rejected criticism of Captain Cook, who charted the east coast of Australia 250 years ago.
"Cook was no slave-trader."
Dr Barker said he was not surprised by Mr Morrison's comments.
"The Morrison government is a conservative government, you would find most members of the party and indeed a lot of people in their supporter base would argue that Cook was a force of good," he said.
"[But] I think that the average person on the street in Australia is far more aware of the nuanced debate about Cook. I find Scott Morrison's line very simplistic, and that we can have a complex debate about our national history."
Additional reporting: AAP