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 has been on and off of the Texans practice squad.
Dysert has yet to appear in a regular season NFL game.
Rivers, 25, originally signed on with the Rams as an undrafted free agent out of Mississippi back in 2013. Since then, Rivers has played for a number of teams including the Jaguars, Dolphins, Broncos and most recently for the Giants.
He has recently worked out for the Saints, Eagles, and Raiders, but did not receive an offer from any of them.
Rivers hasn’t appeared in an NFL game since the 2013 season when he was playing for the Jaguars.