The Cardinals are down to one player on their physically unable to perform list.
Clay signed with the Cardinals in February after being released by the Bills and he was out of practice due to a knee issue. The veteran joins Maxx Williams, Ricky Seals-Jones and Darrell Daniels at tight end in Arizona.
Williams has been sidelined by a back injury. The 2016 third-round pick had 10 tackles while appearing in all 16 games last season.
The Cardinals also announced the signing of defensive end Pasoni Tasini. The move comes a day after the Cardinals cut Darius Philon in the wake of an arrest for aggravated assault with a deadly weapon.