Webb visited the Carolina Panthers and Minnesota Vikings last week, but settled on the team he worked out for in March.
The 25-year-old tackle spent 2013 with the Vikings after being cut by the Chicago Bears. Webb played just 108 snaps while appearing in eight games in Minnesota, starting one.
The seventh-round pick spent three seasons in Chicago -- the last two as the full-time starter. However, he often struggled to protect Jay Cutler's back. Pro Football Focus ranked Webb the 50th-rated offensive tackle in his last full season.
The Chiefs needed to add line help after losing tackle Branden Albert and guards Jon Asamoah and Geoff Schwartz in free agency.
While Webb lacks NFL starting skills, he can be a swing option and provide depth along Andy Reid's line.
The latest "Around The League Podcast" plays the post-draft version of the game "What's More Likely" and breaks down all the latest news.