Bumbery called the death "sudden and unexpected" and said his wife and family were shocked.
It also said the family would be working closely with the medical examiner and asked for privacy.
Pennsylvania State Police will hold a press conference at 1 p.m. Thursday.
With his powerful, almost four-octave vocal range, Cornell was one of the leading voices of the 1990s grunge movement with Soundgarden, which emerged as one of the biggest bands out of Seattle's emerging music scene, joining the likes of Nirvana, Pearl Jam and Alice in Chains.
The 52-year-old Cornell had just performed with Soundgarden at Detroit's Fox Theatre.More news: US aircraft strikes pro-Assad forces in Syria
Cornell had been touring with Soundgarden, and was tweeting upbeat messages about a sold-out concert the band played in Detroit just hours before his death. The Detroit News cited Freeman in a similar report.
The Wayne County Medical Examiner's office will make an official determination about the cause of death.
A detailed account by a Soundgarden fan published in the Detroit Free Press, describes how Cornell looked weak and had difficulty singing the songs entirely. A family friend came upon Cornell's body when he came to check on him around midnight last night.
They disbanded in 2007 - with their track record lit by three Grammy nominations. Like Nirvana, Soundgarden were first signed to indie label Sub Pop before making the jump to a major, taking their heavy, angsty sound into the mainstream with singles like Black Hole Sun.More news: Telefonica among victims of global ransomware attack
Soundgarden was to headline Friday at Rock on the Range, a three-day concert in Columbus, Ohio.
Two Detroit newspapers citing a police spokesman are reporting that the body of Soundgarden frontman Chris Cornell was found with "a band around his neck".
And true to his ideals rooted in the grunge era, he said he would donate his proceeds from his song The Promise (for the 2017 movie of the same name) to the International Rescue Committee, which helps refugees to rebuild their lives.
In 2012, he and his wife Vicky created a foundation that works with vulnerable children facing poverty and homelessness.More news: Mike Huckabee calls Roger Ailes a 'genius'
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