Reed, who was drafted by the Texans, signed a five-year, $22.5 million deal March 10, 2015. He received $9 million guaranteed.
In the 2017 season opener against the Bears, Reed came up with a game-clinching sack in a 23-17 victory.
NFL Network's Ian Rapoport tweeted that Reed's deal was worth about $2 million.
Reed, like Alford, was cut by the Falcons in a cost-cutting move. He played in all 16 games for the Falcons last season, starting half of them, with 24 tackles and a sack.
Reed is expected to move to outside linebacker in the Cardinals’ 3-4 scheme.
The Cardinals have signed several former Falcons in recent years.
In addition to Alford and Reed, the Cardinals signed former Falcons John Abraham (2013), Sean Weatherspoon (2015) and Corey Peters (2016). Peters played in 15 games and started 15 games last season for the Cardinals.