Bush is ranked as the No. 8 prospect available for the draft by ESPN's Mel Kiper Jr. and the No. 10 prospect by Todd McShay.
Bush, a finalist for the Butkus Award as the country's top linebacker and a junior co-captain, injured his hip in the third quarter of the Wolverines' 62-39 loss to Ohio State on Nov. 24.
"Despite getting treatment every day, I have not been cleared to play in the Peach Bowl due to the hip injury I suffered in the last game," Bush wrote in a statement on Twitter. "Due to this injury, I have thought hard about my future in football. Having been injured, I realize how fleeting this sport can be. It's always been a dream of mine to follow in my father's footsteps and play in the NFL.
"This injury has made me realize how important this goal is to me and my family. Therefore I have decided to forgo my senior season and enter the 2019 NFL draft."
His father, Devin Bush Sr., was a safety on Florida State's 1993 national championship team and a first-round pick of the Atlanta Falcons in the 1995 NFL draft. He played eight seasons with the Falcons, Rams and Browns.
Bush had 79 tackles, nine tackles for loss and five sacks for the Wolverines this past season.
The No. 7 Wolverines will be without two of their defensive stars when they play No. 10 Florida in the Chick-fil-A Bowl at Mercedes-Benz Stadium in Atlanta on Dec. 29 (noon ET on ESPN). Junior defensive end Rashan Gary, another projected first-round pick, announced last month that he's skipping the bowl game to prepare for the draft.