Al-Sarkhi has been a fierce critic of Prime Minister Nuri al-Maliki and the United States.
Last week, he blamed some of the bloodshed on the country's top Shiite cleric, Ayatollah Ali al-Sistani, who called for Iraqis to take up arms against ISIS extremists.
Helicopters fired a number of times on the Saif Saad neighborhood, not far from the city's famed holy shrines, on Tuesday night, an employee at a shrine and a resident of Karbala said. The aircraft again hovered over the area Wednesday, as the city was placed on high alert and the streets remained empty of civilians.
Videos posted on YouTube purported to show a firefight between Iraqi forces and the cleric's followers, including one that showed an Iraq Humvee burning as gunfire is heard in the distance.
CNN cannot confirm the authenticity of the videos.
Fighting in Diyala, Salaheddin provinces
Iraqi security forces, with the help of volunteer fighters and Sunni tribesmen, were carrying out an offensive Thursday against ISIS fighters north of Baquba in Diyala province, security officials told CNN on Thursday.
The towns of al-Dawaleeb and al-Shohani, about 35 kilometers (22 miles) north of Baghdad, have been cleared of militants, the officials said.
They said a number of ISIS fighters were killed and wounded during the operation, which lasted 7 hours.
The officials say a large number of bombs were planted in both towns by ISIS, and Iraqi forces were working to defuse them.
Meanwhile, state-run Iraqi TV reported that a counterterrorism unit killed 88 extremists and destroyed 12 vehicles in Salaheddin province Thursday. It offered no evidence of the killings, and no further details were provided.
CNN cannot independently confirm the claims.