The pair spoke out on the issue at the Venice film festival, where they are promoting Clooney's "Suburbicon", a drama that deals with segregation in 1950s America and stars Moore.
They said they backed moves such as a ban on the Confederate flag being hung on public buildings and the rebranding of schools named for heroes of the losing side in the 1861-65 Civil War.
Moore said statues of the Old South's generals and the like "must be removed".
The actress has personally been involved in a campaign to have the J.E.B Stuart High School in Virginia renamed. She attended the school, which is named afer a Confederate general.
"There is a group of people who went to that school who feel now as adults that we need to change this," Moore said.
"The children can't change it but we can and the fact that we take this action means that my children will be in a better place."
"You got to pick if you wanted to be a Union or a rebel soldier, and I was like you want to be the rebel', it was fun.
"You didn't really understand the history of the Confederate flag. That as a flag it was designed to be carried into battle against the United States of America in favour of slavery, and they lost.
"Now if you want to wear it on your t-shirt or hang if off your front lawn, have at it! Good luck with your neighbours.
Americans are deeply divided on the issue. Those who want such statues removed say they amount to a celebration of individuals who supported slavery and are a focal point for a new generation of white supremacists.
Others argue they should stay in place, either because they see the figures commemorated as part of their cultural heritage or because they feel the fact they were put up in the first place reflects US history as it was lived.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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