Ian Rapoport and Tom Pelissero of NFL Network first reported the news.
Veach, who has been with the Chiefs since 2013, will replace former GM John Dorsey, who was fired in June after four years with the team.
According to Terez A. Paylor of the Kansas City Star, Dorsey was dismissed because of "concerns about his internal communication and management styles."
After the team finished 2-14 in 2012, Dorsey and head coach Andy Reid helped lead the Chiefs to four straight winning seasons and a 43-21 overall record.
Veach was also a part of that success. He worked as the team's pro and college personnel analyst for two years before being promoted to co-director of player personnel.
Veach spent his first six seasons in the league with the Philadelphia Eagles, also working with Reid.
"Brett has a sharp football mind, a tremendous work ethic and a keen eye for finding talent," Chiefs chairman and CEO Clark Hunt said, per the team. "Over the last four seasons he's played a critical role in building our football team. I look forward to working with him to continue to build on the strong foundation we have in place."
Veach will now try to get the squad over the top. It's lost in the divisional round of the playoffs in each of the last two seasons.