Kansas City used a fifth-round supplemental pick last year to take UCLA's Justin Medlock, but gave up on him in September. The Chiefs then turned to Dave Rayner, who signed before the season after he was released by Green Bay. Rayner struggled and was released.
Veteran John Carney played in the Chiefs' final five games, going 3-for-3 on field-goal attempts. Cundiff is expected to compete with Carney and Nick Novak, who has played 16 games in the NFL and also signed with the Chiefs on Monday.
Cundiff signed as a rookie free agent with Dallas in 2002 and spent four seasons with the Cowboys. He played for New Orleans in 2006 and was waived by Atlanta before last season after attending training camp with the Falcons.