Butler was signed by Arizona on July 27, just two days before training camp began, and was competing for a backup role behind Kevin Minter.
Clemons, who re-signed with the Cardinals in February, was transitioning from safety to back up Deone Bucannon at dollar linebacker, a hybrid role that Bucannon began playing as a rookie in 2014. Clemons, who had an offseason run-in with police, was put on injured reserve last season with a hamstring injury on Oct. 14, then released five days later.
Butler was released by the San Diego Chargers on March 3 after signing a seven-year, $51.8 million contract extension with San Diego in 2014. He was scheduled to earn $4.65 million in base salary in 2016, and his contract carried a cap hit of $9.28 million for the Chargers. His contract was scheduled to run through the 2020 season.
He was re-signed on Dec. 22, and signed a one-year contract in February.
NFL teams have until Saturday at 4 p.m. ET to trim their rosters to the 53-man limit.