Arizona also re-signed DL Pasoni Tasini to the 90-man roster.
Clay, 30, is a former sixth-round pick of the Dolphins back in 2011. After four years in Miami, the Bills signed him away from the Dolphins and gave him a five-year, $38 million contract that included $20 million guaranteed.
Clay was set to make a base salary of $4.5 million for the 2019 season when the Bills released him. He signed a one-year, $3.25 million deal with the Cardinals this offseason.
In 2018, Clay appeared in 13 games for the Bills and caught 21 passes for 184 yards receiving and no touchdowns. Pro Football Focus has him rated as the No. 50 tight end out of 70 qualifying players.