The 33-year-old started 13 games last season, his first with the Rams.
Matthews missed three games in October after suffering a broken jaw against the Seattle Seahawks on Oct. 3. Overall, he recorded eight sacks, two forced fumbles and 37 tackles (27 solo) in L.A.
The Rams also announced the release of running back Todd Gurley on Thursday afternoon. The two-time All-Pro rusher was owed $10.5 million guaranteed if he remained on the roster beyond 4 p.m. ET. L.A. opted to save the money and part ways after being unable to find a trade partner.
The Green Bay Packers drafted Matthews 26th overall in the 2009 NFL draft. The USC product played for the Pack through the 2018 season, earning an All-Pro honor in 2010, en route to Green Bay winning Super Bowl XLV, and six Pro Bowl nods.
Matthews left Green Bay during last year's free agency period but not before becoming the franchise's all-time leader in career sacks with 83.5.
ESPN's Rob Demovsky explored the possibility of Matthews coming back to Green Bay:
"Before Clay Matthews left Green Bay in free agency last year, he told me he fully appreciated that he would always be primarily associated with the Packers. With the Rams releasing him, a return would make sense -- if he were willing to move to back into inside linebacker. He excelled in that role (even if he didn't love it) when he moved there halfway through 2014 and into 2015 even though he preferred to play outside."
Though SB Nation's Peter Bukowski doesn't foresee a reunion happening:
Matthews' link to his former Packers head coach Mike McCarthy, now the head coach for the Dallas Cowboys, is also intriguing.
Matthews was owed a $2 million bonus if he stayed a Ram beyond 4 p.m. ET Thursday. By cutting him, L.A. saved $3.75 million in cap space, per Rams Wire's Cameron DaSilva.