The man was arrested on an unrelated parole violation June 14, nine days after Elizabeth Smart was reportedly taken at gunpoint from her bedroom in an affluent Salt Lake City neighborhood.
Police Chief Rick Dinse told KSTU television on Sunday that investigators haven't been able to pin down the man's whereabouts on the morning of the abduction. Detectives have been questioning the man for more than a week; he has not been charged in the kidnapping.
"He is potentially a big piece of the puzzle," Dinse told KSTU. "Just how big, we're not sure at this point.
"We haven't been able to satisfy ourselves that he was not involved."
Chris Thomas, who has been acting as a spokesman for the girl's family, said the man last worked in the Smart's Federal Heights home more than a year ago, painting and doing handyman work.
Authorities scheduled a news conference for Monday in Salt Lake City about their interview of Bret Michael Edmunds, whom they had sought for two weeks for questioning in the abduction investigation.
Edmunds was interviewed at a West Virginia hospital by FBI agents and Salt Lake City police detectives.
"I can confirm they have spoken to him. I don't have any details about the context of questioning or the conversations," police spokesman Fred Louis said.
Investigators were analyzing the interviews and continued to follow leads in the case Sunday, but Louis said there were no developments.
"We're still going all day every day, seven days a week," he said.
FBI spokesman Kevin Eaton didn't return messages left by The Associated Press. He said Saturday that questioning Edmunds, 26, likely would not substantially change the Smart case because he was a possible witness and not a suspect.
A milkman helped police trace a car owned by Edmunds that was spotted in the family's neighborhood two mornings before her June 5 abduction.
The car was seen in a suburban Salt Lake subdivision in the days following the girl's disappearance, but search teams recovered nothing Saturday after dogs had picked up a possible scent in the same area one day earlier.
Edmunds checked himself into a hospital in Martinsburg, W.Va., on Thursday for treatment of drug-related liver damage. A nursing supervisor said he remained in serious condition Sunday in the critical care unit.
Authorities said Edmunds was cooperative with investigators and allowed a search of his vehicle.
AP-ES-06-24-02 0229EDT