Sources told ESPN NFL Insider Adam Schefter that Osweiler is not even expected to play in Saturday's game against the Tampa Bay Buccaneers.
Jackson said he informed the team Wednesday of his decision to start Kizer against the Bucs.
"[Kizer] has made a lot of progress by investing the time necessary to learn our offense, working hard to improve on his fundamentals while also effectively moving the offense in preseason games," Jackson said in a statement released by the team. "Development is so important for a young quarterback, this is the next step he needs to take and he deserves this opportunity.
"We are very excited about seeing DeShone in the role of starter for this week, as he is certainly positioning himself well to earn the starting job heading into the regular season."
Osweiler was given the starting job for the two preseason games and produced no points in five drives, completing 12 of 22 passes for 67 yards, an average of three yards per attempt. Kizer, meanwhile, went 19-for-31 for 258 yards with one touchdown and five sacks in two games. The second-round pick from Notre Dame guided nine drives and scored 23 points, all against backup defenders.
"Much like DeShone, Brock, Cody [Kessler] and Kevin [Hogan] have worked extremely hard throughout training camp and have helped create a very supportive environment in our quarterback room," Jackson said. "They've pushed each other, worked well together and we expect that to continue."
In other moves, the Browns placed linebacker Tank Carder (knee) and offensive lineman Matt McCants (ankle) on injured reserve and waived defensve back Justin Currie (ankle) with an injury designation.
Carder, who was competing for a starting role and had started both preseason games, suffered a torn ACL in the first quarter of Monday's game against the Giants and will need season-ending surgery.