They fired live ammunition and tear gas as they began removing tents and concrete barriers near Tahrir Square and a bridge across the Tigris river, eyewitnesses say.
Several people are reported to have been injured in Saturday’s clashes.
Protesters have for months held anti-government demonstrations and camped in the capital.
Saturday’s violence comes a day after a separate massive rally in Baghdad against the presence of US forces in the country.
Many protesters in that rally were supporters of powerful Iraqi Shia cleric Moqtada al-Sadr, who had called a million of people to join Friday’s march.
The US killing of the top Iranian military commander, Gen Qasem Soleimani, on 3 January at Baghdad airport has fuelled tensions.
Also assassinated in the US drone strike was Abu Mahdi al-Muhandis, an Iraqi who had commanded the Iranian-backed Kataib Hezbollah militia group.
Iran responded on 8 January to Gen Soleimani’s assassination by carrying out a ballistic missile attack on two airbases housing US forces in Iraq.
A Pentagon spokesman has said that 34 US service members had suffered traumatic brain injuries.
Hours after the strike, Iran’s armed forces fired two missiles at a Ukrainian passenger plane over Iran’s capital, Tehran, by mistake, killing all 176 people on board.