"He has not confessed to the authorities. He, he, he is not cooperating, but all the people around him are cooperating. And I think that, that, that is very important," Cox told ABC News.
Authorities have identified the alleged killer of Kirk as 22-year-old Tyler Robinson , who is in custody.
Charges against him are expected to be formally filed on Tuesday, Cox said.
He also confirmed a report by the NY Times that the alleged gunman had been communicating with other people via Discord after the shooting.
The Times reported that Robinson made jokes about being the alleged attacker.
"All we can confirm is that those conversations were definitely held and they didn't believe it was really him. It was all, it was all in jest until, until he, you know, until he admitted it was really him," Cox said.

Utah's governor yesterday confirmed reports that the suspect's roommate is undergoing a transition from male to female.
"We can confirm that his roommate was indeed a male-to-female transitioning boy. That's the information that the FBI mentioned yesterday. And we can confirm that as well. And he, he's also cooperating with the authorities," he said.
However, officials have not yet identified a motive for the murder.

Cox said new findings are expected in the coming days and called on Americans to choose kindness at a time of high political tensions.
Before the arrest, CNN previously reported, while police were releasing images of the suspected shooter, Robinson's father became convinced they had shown a picture of his son and confronted him.
"Tyler, is this you? This looks like you," he asked his son, a law enforcement official familiar with the investigation told CNN.
Tyler admitted it was him, the source said, and when his father urged him to turn himself in, he said he'd rather kill himself.
However, his father then convinced him to talk to the youth pastor and Robinson agreed to be detained.