"This thing's open. It really is," Jackson told reporters Saturday at rookie minicamp. "But Cody's done a great job. That's the reason I brought his name up first. He's really improved. He's worked his tail off, and he deserves the right and the opportunity to walk in this building and walk out there first. And they've got to take it from him."
Kessler started eight games last season for the Browns, passing for 1,380 yards with six touchdowns and two interceptions as a rookie.
The Browns acquired Osweiler from the Houston Texans in March, even though he was not expected to be in the team's long-term plans.
"He's competing," Jackson said of Osweiler. "He's here. And I said from the beginning, if a guy is in our locker room we're going to treat him like any of our other players.
"I think in this league we all know you can't have enough good quarterbacks, enough guys to train at the position. You never know how it's going to unfold, and things do happen, but he's competing. He's done a good job. He's been great in the room with the guys. He's been a good person in the building. We're going to continue to allow him to do that and see what he has to show for us and kind of go from there."