The 60-year-old Kelly has coached the Fighting Irish since 2010. Earlier this season, he passed College Football Hall of Fame member Knute Rockne for the most coaching wins in school history.
Kelly guided Notre Dame to the 2013 BCS National Championship Game and College Football Playoff appearances during the 2018 and 2020 seasons.
The Irish currently are ranked No. 6 in the latest College Football Playoff rankings and moved to 11-1 this season after Saturday's 45-14 victory over Stanford.
Once the agreement is finalized, Kelly will replace Ed Orgeron as coach of the Tigers. LSU announced in October that Orgeron wouldn't return as the school's head coach next season, less than two years removed from an undefeated season and national title victory.
In the two seasons since that championship, Orgeron led the Tigers to an 11-11 combined record and was accused of improperly handling accusations of sexual misconduct by a player twice in the last year.
LSU earned an upset win over Texas A&M on Saturday for its sixth victory of the season. After the win, Orgeron said he won't be on the sideline in the Tigers' upcoming bowl game, with offensive line coach Brad Davis to serve as the team's interim head coach.
Kelly previously served as the head coach at Grand Valley State (1991-2003), Central Michigan (2004-06) and Cincinnati (2006-09). He compiled a 92-39 record over 12 seasons at Notre Dame.