Cornell's body will be flown from Michigan to Los Angeles on Sunday, and Cornell will be buried at the Hollywood Forever Cemetery, the sources said.
One of the sources told CNN, "The family is thinking about a (public) memorial for fans, but is coping now with their loss and the funeral service."
Cornell, 52, died Wednesday night after performing with Soundgarden at the Fox Theatre in Detroit.
The medical examiner's office in Wayne County, Michigan, ruled the death a suicide by hanging.
In a statement to CNN, Cornell's widow, Vicky Cornell, said she became concerned for her husband's well-being after he told her he took "an extra Ativan or two."
Ativan is a brand name for lorazepam, a prescription drug used to treat anxiety.
Vicky Cornell said she became concerned and had her husband's personal security team go to his MGM Grand Detroit hotel room. They broke down the door and found Cornell unresponsive on the bathroom floor in his room.
Cornell's widow said in her statement that she did not believe he "intentionally" committed suicide
. The family's attorney, Kirk Pasich, said in a statement that the family would await the toxicology results to see if Ativan or other drugs contributed to his death.
"The family believes that if Chris took his life, he did not know what he was doing, and that drugs or other substances may have affected his actions," Pasich said.
Cornell began his career in the Seattle rock scene. In 1984, he formed the band Soundgarden
, which was influential in the grunge-rock movement of the early 1990s.
The group hit its commercial peak in 1994 with its breakout album, "Superunknown," which was nominated for a Grammy Award for best rock album. A hit song from that album, "Black Hole Sun," won the Grammy that year for best hard rock performance and the song "Spoonman" won for best metal performance.
Soundgarden broke up in 1997. In 2001, Cornell joined with three former members of Rage Against the Machine to form Audioslave.
Soundgarden returned to the stage in 2010 for festival performances and had recently started touring again.