Staley had written drug-related lyrics
The 34-year-old singer and guitarist was found dead on Friday at his flat in Seattle.
Authorities said he had been dead for two weeks before a relative discovered him.
"There was nothing suspicious about the death. It appears to be an overdose or possibly a natural death," Seattle Police spokesman Duane Fish said.
A post-mortem was carried out on Saturday, but the cause of death will not be confirmed for weeks while toxin tests were being conducted.
Because of the decomposition, tests had been needed to confirm the identity of the body.
Fans have been leaving flowers on the steps of the building where Staley lived, and more than a hundred held a candlelit vigil by the Seattle Center.
Grunge heavyweights
His death comes eight years after Kurt Cobain, leader of Seattle's best-known rock band Nirvana, killed himself, with heroin later found in his bloodstream.
Alice in Chains were founded in 1987 and ranked with Nirvana, Pearl Jam and Soundgarden as one of the bands that made Seattle famous in the early 1990s.
They recorded seven albums, including a Billboard number one in 1995.
The band stopped touring in the mid-1990s when Staley went into rehabilitation for heroin use.
He told Rolling Stone magazine that lyrics about heroin use came from his own experience.
"They worked for me for years, and now they're turning against me - and now I'm walking through hell, and this sucks," he said in 1996.
But he had maintained he would never think of committing suicide.
I'm scared of death, especially death by my own hand," he said.
"At the end of the day or at the end of the party, when everyone goes home, you're stuck with yourself."