Campaigners have been calling for the removal of controversial statues across the country including monuments in Edinburgh to former home secretary Henry Dundas and the 2nd Viscount Melville Robert Dundas, and in Glasgow to the likes of historian Thomas Caryle in Kelvingrove Park and Robert Peel in George Square.
It comes after a statue of slave trader Edward Colston was toppled and thrown into a river in Bristol during the Black Lives Matter protest, which was sparked by the death of George Flloyd in Minneapolis in May.
Police Scotland's Assistant Chief Constable, Kenny MacDonald, said they are aware of a "small number" of statues being vandalised recently and that extra patrols around such monuments and statues will be in place in the coming days.
He said: "We are aware of a small number of events scheduled to take place this weekend, however our firm advice remains that people should find alternative ways to protest rather than attending mass gatherings, for example through digital means.
"I am aware that vandalism of a small number of monuments and statues has taken place in recent days. Such action is criminal and will be fully investigated. Police officers will be providing additional patrolling around such monuments and statues in the coming days to deter such criminality.
"We understand that people want to make their voices heard, but they must do so lawfully and peacefully.
"We would encourage everyone to continue to follow the Scottish Government's regulations and guidance to help the collective effort to protect the NHS and save lives.
"There will be an appropriate policing plan in place for the coming days and into the weekend."
Anti-racism protesters gathered at sites across the nation - including Holyrood Park and Glasgow Green - at the weekend in the wake of the death of George Floyd.
George Floyd died in Minneapolis in May as a white police officer held a knee on his neck for nearly nine minutes.
Most people involved in the protests have been wearing face coverings and all who attend are asked to maintain social distancing.
Politicians have also urged people to avoid mass gatherings and adhere to two-metre social distancing rules in public places.