Instead, they want the suspect to be offered life in prison without parole, the Sandusky (Ohio) Register reported Wednesday.
The trial of Curtis Clinton, 42, is scheduled to begin Monday. He is charged with the strangulation deaths of Heather Jackson, 23, and her children, Celina, 3, and Wayne IV, 20 months, in September 2012.
Heather's mother, Jody Lynch, and Lynch's husband, Herbert, as well as Heather's brother, Nick Fee, say they have told Erie County Prosecuting Attorney Kevin Baxter they want him to offer Clinton a life sentence to avoid a lengthy trial.
The family has attended every court hearing on the case.
Jody Lynch said she wanted Clinton to go to jail.
"I know he won't make it in there," she said. "And then he'll suffer like my kids suffered, instead of going into a room and getting a shot, living like us."
Baxter has not responded to the family but has said in a statement he had "seriously considered" the request.
"Taking this into consideration, as well as other aggravating factors, the state is moving forward with the trial," Baxter said.