At present Stephens' claims of other killings can not be substantiated, but the man he shot on social media has been named as 74-year-old Robert Godwin Sr. The murder of 74-year-old Robert Goodwin Jr. has set a manhunt for Stephens underway.
Police said they have searched "dozens of locations" for the suspect, Steve Stephens, and tried to convince him to turn himself in when they spoke with him on his cell phone on Sunday.
Police Chief Calvin Williams warned residents to be careful as the go about their day.
The suspect, Steve Stephens, also claimed to have committed multiple other homicides.
Police crews continue to search multiple areas in the city, but residents of Pennsylvania, New York, Indiana and MI are urged to be on alert, as Stephens may have fled the state.
"The only way to help is to give us information to bring him in peacefully, safely", said Mr Williams.
Police said they have talked with family and friends of Stephens, who is a case manager at Beech Brook, a behavioral health agency headquartered in Pepper Pike, near Cleveland.
Williams said Stephens may be driving a newer model white Ford Fusion with temporary OH license plate E363630. In January, four people in Chicago were arrested after they allegedly beat and taunted a mentally disabled man live on the social media site.
A daughter of the elderly Cleveland man shot and killed in a gruesome video posted on Facebook says he was a gentle person, with nothing mean about him. Stephens' Facebook page has also been removed.
Stephens had just recently broken up with his girlfriend of three years, who he named as "Joy Lane" and says the attacks are because of her. They said Stephens and Godwin didn't know each other.
Beech Brook spokeswoman Nancy Kortemeyer says Monday there was nothing in the background check for 37-year-old Steve Stephens that caused any concerns.
Whereabouts: The Dayton Daily News reports that Stephens was last seen south of Interstate 90 in the area of East 93rd in Cleveland.
On Sunday, Green learned about the shooting when her youngest son told her about the video. First get away from him as quickly as you can, and then when you're sure you're safe, call the police. "We are horrified. We are hoping for a quick resolution and we hope no one else is harmed". Authorities have warned people in Pennsylvania, New York, Indiana and MI to be alert for him, and the FBI is assisting in the investigation. He is described as a black man with a bald head and beard, standing 6 foot 1 inch and weighing 240 pounds.
Authorities consider Stephens very unsafe and ask anyone with any information about his whereabouts to call 911 immediately.