Media playback is unsupported on your deviceMedia captionThe suspect said he was looking for victims in a video posted on Facebook Investigators in Ohio say a gunman who shot a grandfather on the street apparently at random and posted the footage on Facebook could be anywhere.
The suspect, Steve Stephens, said in another video post that he had killed 13 people and would kill more.
He shot dead 74-year-old Robert Godwin as the victim reportedly walked home from Easter lunch on Sunday afternoon.
Police have asked residents of Ohio, Indiana, Michigan, New York and Pennsylvania to be on the alert.
A ‘monster’ who ‘just snapped’
Cleveland police department have issued a photo of Mr Stephens, 37, on its website, describing him as a black male, 6ft 1in (1.9m) and 244lb (110kg).
Image copyright Twitter Image caption Mr Godwin was a grandfather and father of nine
He is thought to be driving a white Ford Fusion, but may have switched cars.
They said he was probably still armed and dangerous.
Cleveland Police Chief Calvin Williams said in a press conference on Monday morning: “Steve is still out there some place… we’re still asking Steve to turn himself in, but if he doesn’t we’ll find him.”
FBI Special Agent Stephen Anthony said: “Quite frankly, he could be in a lot of places.”
Image copyright Facebook Image caption Steve Stephens claimed to have killed more than a dozen others
US Marshall Peter Elliot said law enforcement agencies across the US were helping create a dragnet to “make this individual’s world very, very, very small”.
Investigators said Mr Stephens’ boast about other killings was still not verified.
Police say the victim in Sunday’s graphic video appeared to have been selected at random.
The suspect approaches Mr Godwin and asks him to say the name of a woman believed to be the gunman’s former girlfriend, before shooting him in the head.
Image copyright Facebook Image caption The old man appears to attempt to shield himself with a plastic bag
“Can you do me a favour?” asks the gunman. “Can you say Joy Lane?”
The victim says: “Joy Lane.”
Mr Stephen says: “Yeah, she’s the reason why all this about to happen to you. How old are you?”
The grandfather and father of nine appears to try to shield himself by holding up a plastic bag.
Image copyright Cleveland Police Image caption Police issued a photo of the suspect
The woman, Joy Lane, confirmed to CBS News that she and Mr Stephen had been in a “relationship for several years”.
“I am sorry that all of this has happened. My heart & prayers goes out to the family members of the victim(s). Steve really is a nice guy… he is generous with everyone he knows.
“He was kind and loving to me and my children,” she wrote in a text message.
Maggie Green, the suspect’s mother, told CNN that he was “mad with his girlfriend. That’s why he is shooting people and he won’t stop until his mother or girlfriend tell him to stop.”
Image copyright Twitter Image caption Police believe the suspect’s car may have been newly purchased
Mr Stephens is employed by Beech Brook, a children’s behavioural health agency, according to a company spokesman.
Facebook said in a statement on Sunday that it co-operates fully with the authorities in incidents where there are “direct threats to physical safety”.
Responding to the reports involving Mr Stephens, the company said that it was “a horrific crime”, adding that it “does not allow this kind of content” on its site.