“These are markers to a person’s grave,” said David Sloane, a historian at the University of Southern California who has written two books on cemeteries. “Cemetery memorials do have a different meaning than a symbolic public memorial on the highways and byways of the city or in a public park.”

The monument targeted for removal, boxy and carved from a smooth gray granite, is engraved with the names of dozens of soldiers, mostly men who were imprisoned and died at nearby Camp Randall during the Civil War. It stands prominently in front of the men’s graves, their names chiseled on their headstones in simple block letters — C. A. Hollingsworth, H. Faulks and L. Galloway among them — alongside their regiments and home states, frequently Alabama, Tennessee and Mississippi. (Those who favor removing the monument say they have no intention of altering the gravestones.)

Three separate city council committees intend to study the memorial, which was installed by the United Daughters of the Confederacy around 1931 and also honors a local woman who regularly tended the graves, and make recommendations on what to do with it — whether to alter the structure, remove it entirely or append more information to it to give visitors greater context. Ms. Rummel said she favored its removal, reasoning that the cemetery “is not a town square, but it is a public space.”

Mr. Soglin initially believed that it should be removed, but said he has changed his mind.

“The more we’ve had this discussion and the more ignorance I witness in regard to this period of our history, the more I’m convinced it should be left and a plaque giving an accurate description of this era of the black codes, of Jim Crow, be told,” he said. “We want to educate Americans about the Civil War and its aftermath.”

On a quiet, tree-lined street of houses that borders Forest Hill Cemetery, residents said they had been mulling the issue.

Standing in the backyard of a two-story Tudor where he has lived for 30 years, Rod McKenzie, a retired engineer, pointed over his fence to the grassy lawn of Forest Hill, lined with small grave markers.