When detectives asked Balfour to take the polygraph test, he stopped cooperating and refused to take the test, said the official, who is familiar with the investigation.
Balfour has been at the center of the investigation since shortly after the bodies of Hudson's mother, 57-year-old Darnell Donerson, and brother, 29-year-old Jason Hudson, were found Friday inside their home on Chicago's South Side. The body of Balfour's stepson, 7-year-old Julian King, was discovered Monday in an SUV on the city's West Side. All three had been shot.
Meanwhile, a funeral home announced Thursday that services for Hudson's mother, brother and nephew will be held on Monday morning. The service at Apostolic Church of God on Chicago's South Side will be closed to the public.
Balfour had not been charged as of Thursday. Investigators were still gathering evidence and following up on leads, the official said. However, the official said police do not believe anyone else was involved in the killings.
It is possible that someone else drove the SUV with the boy's body inside and parked it on the West Side, the official said, but Balfour is "the only suspect in the killings (and there is) nothing to link a second individual" to them.
A police spokeswoman declined to comment on the official's statements.
Balfour's mother, Michele Balfour, has said Hudson's mother kicked Balfour out of the family home last winter. She denied her son had anything to do with the killings.
It was unclear whether Balfour had an attorney.
On Wednesday, police found a gun in a vacant lot around the corner from where the SUV was parked.
"There was also some small bits of evidence recovered," Deputy Chief Nicholas Roti said. He did not elaborate and declined to take questions.
The gun, discovered in thick shrubs, was sent to the Illinois State Police crime lab for testing, Roti said.
Records show that Balfour was arrested with cocaine in his car in June, but authorities declined to return him to prison on a parole violation.
Corrections Department spokesman Derek Schnapp said officials who reviewed the cocaine-possession case against Balfour determined "the evidence that was presented during that time wouldn't have necessarily warranted a violation."
A judge dismissed the charge in July for lack of probable cause, but under the strict rules of the state's parole program, Balfour could have gone back to prison just for the arrest.
Balfour served seven years for a 1999 attempted murder and vehicular hijacking conviction.
The Illinois Department of Corrections issued a warrant for Balfour on Saturday for violating terms of his parole by possessing a weapon and failing to attend anger management counseling and a substance abuse program, according to his parole history report.
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