Capt. Lisa M. Rudzinski, commander of the London, Ky., post, says there will be a resolution in the case probably within weeks, not months, the Louisville (Ky.) Courier-Journal reports.
"We're much closer than we were," Rudzinski said Tuesday. "I think we are toward the end."
She said investigators are currently awaiting the results of forensic tests from the state crime lab.
Sparkman, 51, was found gagged and tied to a tree last month near a remote Clay County cemetery.
He was naked and the word "fed" had been scrawled on his chest.
Police have not categorized his death as a homicide, suicide or accident.
"The mere fact that we have not made a determination after such a time lapse is unusual but not something that's never happened," Rudzinski said.