The injury occurred when Jenner was hit by a puck during practice on Sunday, according to the release.
Jenner recorded 16 goals and 33 assists as a rookie last season. He was drafted in the second round of the 2011 draft.
The Blue Jackets finished 43-32-7 last season before being eliminated in the first round of the playoffs by the Pittsburgh Penguins.
Columbus opens the upcoming season Oct. 9 against the Buffalo Sabres.
- Chris Johnson