Browns head coach Hue Jackson announced earlier Wednesday that rookie second-round pick DeShone Kizer will get the start under center.
Tom Pelissero of NFL.com reported the Browns "continued to make calls" about a potential trade of Osweiler before and after the start of training camp. Pelissero noted Cleveland is "keeping [its] options open."
Osweiler is part of a three-way quarterback competition that includes Kizer and second-year man Cody Kessler. Of the three, Osweiler has had the least success during the preseason.
In two games, Osweiler has completed 54.5 percent of his passes for 67 yards, no touchdowns and one interception.
After he threw for 2,957 yards with 15 touchdowns and 16 interceptions in 2016, however, the Texans traded the 26-year-old to Cleveland.
The deal was essentially a salary dump on Houston's part, as it also traded a 2018 second-round draft pick to the Browns as part of the transaction.
Osweiler has failed to recapture the promise he once showed during his time with the Denver Broncos, while Kizer has impressed with 258 passing yards, one passing touchdown, 47 rushing yards and one rushing score in two preseason contests.