Mora would replace Rick Neuheisel. He was fired after the Bruins went 6-6 this year in the regular season, including a 50-0 loss against USC.
Overall Neuheisel went 21-29 over four seasons at his alma mater.
Mora is a former Atlanta Falcons and Seattle Seahawks coach, going 31-33 over four seasons.
Mora is the first football coach of the Bruins in more than 60 years with no ties to the school. He will be introduced at a news conference next week.
UCLA will play Illinois in the Kraft Fight Hunger Bowl on Dec. 31 under interim coach Mike Johnson, Neuheisel's offensive coordinator. Johnson also was Mora's quarterbacks coach for two seasons with the Falcons.
The Bruins haven't played in a Rose Bowl in 13 seasons and haven't been ranked for more than five seasons- both school-record streaks.
The Associated Press contributed to this story.