Young was not in the league last season, and general manager Ray Farmer said he wants to see him perform and that the minicamp tryout is the best way.
"He's had success in the National Football League," Farmer said. "We wanted to take the opportunity to get the chance to see where he was. We'd been kind of following him through the offseason. He'd been doing some things that kind of caught our eye and made us kind of take note."
The Browns need to add a veteran to back up Brian Hoyer, who is penciled in as the starter heading into this season but is coming off knee surgery. Other than Hoyer, the only quarterback on Cleveland's roster is Alex Tanney, who was signed late last season as the Browns were hit hard by injuries at quarterback.
Young last played in the NFL for the Eagles in 2011, though he was with the Packers during the preseason in 2013. Thigpen played in four games for Buffalo in 2012.
Both fit the play-action rollout system that offensive coordinator Kyle Shanahan has run.
Thigpen has played in 29 games in six seasons, completing 54 percent of his passes with 21 touchdowns and 18 interceptions.
Young made 50 starts between 2006 and 2011. He completed 57.9 percent of his passes with 46 touchdowns and 51 interceptions. He is 31-19 as a starter and twice made the Pro Bowl.
The Associated Press contributed to this report.