Dysert, 25, is a former seventh-round pick of the Broncos back in 2013. He was placed on waivers by the Broncos at the start of the 2014 season, but they later added him to their practice squad.
The Bears claimed Dysert off of waivers a few weeks ago, but he was among their final roster cuts at the start of the season.
Since then, Dysert returned to the Texans, but was on and off of their practice squad before being cut loose a few weeks ago.
Dysert has yet to appear in a regular season NFL game.