Franklin County Circuit Judge Phillip Shepherd took issue with the state's lethal injection process in 2010 and blocked executions going forward.
And that ruling has held for 15 years.
But Kentucky Attorney General Russell Coleman is pressing ahead with efforts to restart the process, filing motions that could potentially lift the 2010 ban.
Meanwhile, State Sen. Brandon Smith, a Republican from Hazard, is asking Beshear to sign a death warrant in the case of Ralph Baze, who was convicted of murdering a Kentucky sheriff and deputy more than three decades ago.
Thursday, Beshear said one part of the court's requirements for reconsidering the ban is in motion.
"One of its holdings was we did not have a regulation that would be necessary before signing any death warrant, so that regulation is going through the process right now," the governor said. "It is filed, it is in the current schedule, and that's the step that had to be taken first under that order. And that's the step that's being taken."
Currently, 24 individuals are on death row in Kentucky.