Australian Prime Minister Malcolm Turnbull has described the moves as a “Stalinist” attempt to rewrite history.
The tensions in the United States over whether or not to remove statues of proslavery Confederate Civil War leaders in Charlottesville, Virginia, has stoked the debate in Australia and elsewhere.
Political leaders in Sydney are mulling over whether or not to alter a statue, erected in 1879, to commemorate Cook.
An engraving on the statue’s base reads: “Discovered this territory, 1770.”
Critics say Aboriginal people native to the continent had known it as their home for around 60,000 years before Cook dropped anchor in Botany Bay.
Sydney City Council has referred the issue, along with a second statue of Governor Lachlan Macquarie, the administrator who turned the British penal colony into a free settlement, to an indigenous advisory board.
Mr Turnbull dismissed the issue as the preoccupation of a Left-wing political fringe.
He said: “We can’t get into this sort of Stalinist exercise of trying to white out or obliterate or blank out parts of our history.
“All of those statues, all of those monuments, are part of our history and we should respect them and preserve them.”
Sydney’s Lord Mayor Clover Moore said yesterday: “I think that’s a conversation that really needs to have federal leadership because it’s about Australia, it’s about who we are.”
Australia’s indigenous people are battling for constitutional recognition and some want to change Australia Day from January 26, which commemorates the day British colonists first settled there.
Meanwhile, New York mayor Bill de Blasio has said he may order the removal of the city’s statue of Christopher Columbus.
The statue was put up in 1892 to commemorate the 400th anniversary of Columbus landing in the Americas.
It has been suggested by local government officials in several American cities that monuments to Columbus should be taken down because of his brutal treatment of indigenous communities.
Italian-American groups are furious at the threat to Columbus, born in Genoa.
A row broke out this week in London over calls to topple the Trafalgar Square landmark Nelson’s Column in the wake of the Charlottesville protests.
Commentator Afua Hirsch branded Admiral Nelson an “unashamed white supremacist”.
That sparked outrage with one critic tweeting: “Britain would not be Britain if not for this man.”