Ian Rapoport of NFL.com first reported the addition Monday.
The New Orleans Saints, San Francisco 49ers and Seattle Seahawks also put in claims, per Rapoport, adding the Baltimore Ravens "may have" signed him if he cleared waivers. Mike Florio of Pro Football Talk reported he planned to report to the Chiefs.
That comes after ESPN's Adam Schefter reported on Saturday that Suggs would "strongly consider not reporting if a team other than the Ravens claims him, league sources say," adding that the Ravens were his "overwhelming preference."
Suggs was enjoying a solid season for the Cards with 37 combined tackles, 5.5 sacks and four forced fumbles in 13 appearances. His release was more about a team looking toward the future after being eliminated from playoff contention than a lack of on-field production.
Arizona head coach Kliff Kingsbury told reporters the organization decided it was better to let the seven-time Pro Bowl selection find a new home than force him into a reduced role behind 2017 first-rounder Haason Reddick:
"We've kind of made the shift to playing some younger players, we've moved Haason out there. In fairness to him, to try and find a better fit right now, we released Terrell today. Just have so much respect for him and what he brought and just wanted to give him that opportunity. He was phenomenal in our building. Work ethic, energy, the juice he brought everyday, professionalism was through the roof."
Suggs is no longer the dominant three-down force he was during his peak with the Baltimore Ravens—he ranks second among all linebackers in approximate value since his rookie season in 2003, per Pro Football Reference—but he's still capable of making a key impact as a situational pass-rusher.
The 37-year-old Arizona State product hasn't seriously contemplated retirement, at least publicly, despite being the third-oldest defensive player in the NFL this year. He signed with Arizona in March after spending his first 16 seasons with the Ravens.
"Change can be scary but also necessary, so it's very exciting," Suggs said after joining the Cardinals, per Scott Bordow of The Athletic. "The possibilities. It's very fun. It's amazing what a breath of fresh air can do to revive you."
The Chiefs give the Minneapolis native a potential chance at winning a title this season, while Suggs would give the team another solid pass-rusher to pursue Lamar Jackson and other quarterbacks in pursuit of a Super Bowl.