SANTA CLARA -- The 49ers looked to upgrade their roster this week with a waiver claim for veteran edge rusher Terrell Suggs, whom the Arizona Cardinals cut on Friday.

But, instead, Suggs will be going to the Kansas City Chiefs, who had higher waiver priorities than the 49ers and three other teams.

“We just thought he would help us, a good pass rusher out there who’s a vet," 49ers coach Kyle Shanahan said on Tuesday. "[Suggs] could help a lot of teams and we would’ve definitely preferred for him to help us over people we have to go against.”

The Seattle Seahawks and New Orleans Saints also put in waiver claims for Suggs. The Baltimore Ravens were last in the waiver priority, and they likely would have pursued signing him as a free agent if he had cleared waivers.

The 49ers are down three edge rushers with Dee Ford out indefinitely with a hamstring injury. In addition, Ronald Blair (knee) and Damontre Moore (forearm) sustained season-ending injuries.

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Suggs, 37, has 138 career sacks over a 17-year career, including 5.5 this season in 13 games with the Cardinals.