His friend Edward Burch told the Chicago Sun-Times that the multi-instrumentalist had died in his sleep and that an autopsy would be performed.
On the group's official website, Wilco frontman Jeff Tweedy wrote that the band was "deeply saddened by this tragedy".
Tweedy said: "We will miss Jay as we remember him - as a truly unique and gifted human being and one who made welcome and significant contributions to the band's songs and evolution.
"Our thoughts go out to his family and friends in this very difficult time."
Bennett, who had been suffering from a painful hip injury before his death, was a member of Wilco from 1994 to 2001 and released five solo albums after his split from the group.
He left Wilco after a series of conflicts with Tweedy that can be seen in the 2002 Sam Jones film I Am Trying to Break Your Heart: A Film About Wilco.
Earlier this month, Bennett sued Tweedy for breach of contract regarding his appearance in the film, in addition to alleged unpaid royalties for his work with the band.