The source said the Steelers will receive a late sixth-round pick in return.
Gilbert appeared in 88 games (87 starts) over eight seasons with the Steelers after they drafted him in the second round in 2011. The 31-year-old is entering the final year of his contract and is set to make $4.9 million in 2019.
He appeared to indicate his time in Pittsburgh was over earlier Friday, when he tweeted "Next chapter."
According to NFL Next Gen Stats, in a limited sample last season (5 starts, 194 pass blocks), Gilbert recorded a pass block win rate of 87 percent -- the best of all Steelers offensive linemen. The Cardinals, meanwhile, were the worst pass-blocking team in the NFL last season, based on win rate.