O'Donnell has spent all five of his seasons with Chicago and is the franchise's leader with a 44.9-yard career gross average. O'Donnell averaged 45 yards per punt last season and tied a career high with 28 punts inside the 20.
The Bears also agreed to a one-year contract with backup quarterback Tyler Bray. He spent most of last season their practice squad.
The Bears announced the moves on Wednesday. Chicago won the NFC North at 12-4 in coach Matt Nagy's first season.