The team said Bush, 52, the assistant general manager for the past five seasons, will take over following the exit of Hendry, 56, who told a Wrigley Field news conference he was informed of the decision July 22.
"It's professional baseball ... . If you don't win enough games, you can't fight change," said Hendry, who became GM in July 2002 and the Cubs made the playoffs in 2003, 2007 and 2008 but are headed toward a second straight losing season.
"We're here to win games, and in the last couple of years we didn't win enough. I will leave here with nothing but gratefulness for being part of this organization for 17 years. Not many people get a chance to do that."
Cubs Chairman Tom Ricketts thanked Hendry for his service.
"My family and I appreciate Jim's dedication during our time with the Cubs and thank him for his overall 17 years of service to the Cubs organization," Ricketts said. "It is time for a fresh approach in our baseball leadership and our search begins immediately for our next general manager."