Reed can earn up to $7 million on the deal, sources said.
Reed joins a defensive tackle rotation that includes Chris Jones, Derrick Nnadi and Tershawn Wharton. Jones has led the Chiefs in sacks in each of the past three seasons. Nnadi plays mostly on running downs, while Wharton showed promise last season as an undrafted rookie.
But Reed, with his 19 sacks over the past three seasons, gives the Chiefs a strong interior pass rusher to combine with Jones.
The Seattle Seahawks released Reed on Friday, a year after the team signed him to a two-year, $23 million extension.
Reed, 28, has been a full-time starter since 2017, the year after Seattle chose him in the second round out of Alabama. He was known primarily as a run-stuffer his first two seasons but then broke out for 10.5 sacks in 2018. He began the 2019 season by serving a six-game suspension for violating the NFL's personal-conduct policy and finished the year with two sacks in 10 games.
He had something of a bounce-back season in 2020 with 6.5 sacks in 16 games but was set to count nearly $14 million against the salary cap before he was released.
For his career, Reed has 22 sacks and 194 tackles in 72 regular-season games.
ESPN's Adam Teicher and Brady Henderson contributed to this report.