Puig, 23 next week, is expected to arrive in the United States in the coming days to close the deal before the MLS transfer window closes on Thursday.
As first reported by Mundo Deportivo, sources expect Puig to join the Galaxy for free, but Barca will be entitled to a percentage of any future transfer.
Barca had hoped to bring in a fee for Puig this summer, but, with just one year to run on his contract, that proved impossible for a player they had made quite clear was not part of their immediate future.
Puig was told by Barca coach Xavi Hernandez in May that he is not part of his plans for the upcoming season. He was left off the club's tour of the United States last month to allow him to resolve his future.
Several European clubs have been linked with Puig but he has been enticed by the option of joining the Galaxy, with former Barca midfielder Victor Vazquez selling the club to him.
Puig joined Barca's academy in 2013 and made his first-team debut in 2018. He has since made 60 appearances for the club, but has never established himself as a regular despite being tipped as one of the pillars of Barca's future when he first broke through.
He featured infrequently under Ernesto Valverde, but was handed more regular game time by Quique Setien. Ronald Koeman, who replaced Setien, advised him to go out on loan to play regularly, but Puig preferred to stay and fight for his place.
The return of Xavi did not change Puig's situation last season and, after playing just 18 times in all competitions in 2021-22, a move elsewhere became inevitable.
The Galaxy are in eighth place in MLS' Western Conference with 30 points from 22 games and one spot outside the playoff places.