"You could have broke the case right then and there," Castro told police during a recorded interview that was obtained by NBC and first reported Friday on the "Today" show.
Cleveland police did not respond to requests for comment regarding Castro's claim that there was a missed opportunity to catch him after DeJesus disappeared.
The recording shows the former school-bus driver eating a slice of pizza and later pacing the room during a reportedly four-hour interrogation in which he told police he had used victim Amanda Berry's cell phone to call her mother and say she was alive.
"I think I said something ... that I have her daughter and that she's OK, and that she's my wife now — something like that, you know, probably not the exact words," he told investigators.
Castro also describes what he considered a close call: a girlfriend spotted a television on in a room occupied by victim Michelle Knight. Castro told police that made him think he might be caught.
Castro, 53, was a month into his life sentence when he hanged himself in his prison cell Tuesday night. A funeral home picked up his body Friday from the Franklin County Coroner's office on behalf of Castro's family.
In the recording of his interrogation following his May arrest, Castro, handcuffed and dressed in dark clothes, is asked about suicidal thoughts. A search of Castro's home had turned up a 2004 confession note in which he wrote he wanted "to put an end" to his life.
"And what about suicide? You still?" an investigator asked him.
"I just want to crash through that window," Castro answered.
Castro committed suicide using a bedsheet despite his placement in protective custody, which involves checks every 30 minutes. He had been taken off suicide watch while in county jail.
The state has launched two probes related to Castro, Ohio prisons spokeswoman JoEllen Smith said. One looks into the suicide, and the other examines whether Castro received proper medical and mental health care leading up to his death.
A spokeswoman for the city of Cleveland and its police department said Castro's case records are being reviewed to determine what will be made public under Ohio law. Maureen Harper said the city isn't the source of the video that aired Friday.
Through a spokesman, Cuyahoga County Prosecutor Timothy McGinty also denied having released the recording. The FBI's Cleveland division spokeswoman said the office did not release the video.
Authorities speculated Castro's defense team released the tape. Attorney Jaye M. Schlachet on Friday declined to comment on the matter.
Representatives of the victims did not respond to messages seeking comment Friday.
Castro was sentenced Aug. 1 to life in prison plus 1,000 years after pleading guilty to 937 counts, including kidnapping and rape, in a deal to avoid the death penalty.
Castro's captives disappeared separately between 2002 and 2004. They were rescued from Castro's house May 6 when Berry broke through a screen door and called 911.
Investigators said the women were bound, repeatedly raped and deprived of food and bathroom facilities.
The dispatcher who took Berry's call was disciplined Friday for failing to stay on the line with her until police arrived.