ESPN's Josina Anderson first reported the two sides were in discussions on a deal.
The 46-year-old's arrival is likely to be met with some consternation among Arizona's fans. The Denver Broncos fired Joseph in December after going 6-10 in 2018. They hadn't posted a losing record in back-to-back years since 1972.
One could argue two seasons isn't a big enough sample by which to definitively judge Joseph's future prospects as an NFL head coach. His firing also papered over larger structural problems that extended beyond the coaching, as The Ringer's Danny Heifetz argued:
"Much of the blame for Denver's past two years goes to Joseph, but a large part is also due to a declining roster that has been mismanaged by John Elway since the team won Super Bowl in 2016. Joseph was rumored to be on the hot seat at the end of last year, but Elway chose to retain him because he felt part of the team's decline was due to a subpar roster (evidenced by the Trevor Siemian–Paxton Lynch decision at quarterback)."
Having said that, the Broncos were justified in letting Joseph go.
The Athletic's Nicki Jhabvala detailed many of his biggest strategic blunders. None was more glaring than attempting a field goal from the Cleveland Browns' 6-yard line on fourth down when trailing 17-13 with five minutes, 29 seconds left in the fourth quarter.
For much of his coaching career, Joseph was a defensive backs coach until the Miami Dolphins made him their defensive coordinator in 2016. He spent only one season with them before the Broncos brought him aboard to succeed Gary Kubiak.
With the Broncos, Joseph simply looked like a coach in over his head, so going back to a coordinator makes sense. He'll be under slightly less scrutiny with Arizona, and he can plot a possible return to the head coaching ranks one day.
The Dolphins showed progress during his only season as the defensive coordinator. Miami ranked 19th in defensive efficiency in 2016, up from 25th the year before, per Football Outsiders.
As underwhelming as Joseph's tenure was in Denver, he's a shrewd addition to the Cardinals' staff to work under the offensive-oriented Kliff Kingsbury.