Clayborn, 30, played for the Falcons from 2015 to '17 and had a career-best 9.5 sacks for Atlanta in 2017.
He became a free agent after the New England Patriots released him on March 15.
He had 2.5 sacks in 2018, his only season with the Patriots, playing 30 percent of the defensive snaps with his primary contributions coming in obvious passing situations. His uncertain future with the team came into focus late in the year when he was a healthy scratch for the final two games. He had signed with them as a free agent.
He has 32.5 sacks, eight forced fumbles and 178 tackles in eight NFL seasons since being selected 20th overall by the Tampa Bay Buccaneers in the 2011 draft.