The five-time Champions League winner released a statement to bid farewell to Welsh fans, saying his decision to retire from international football had been "by far the hardest of my career."
The 32-year-old Bale will join LAFC as a free agent with his contract due to end this summer as the signing is expected to be announced "within a few days."
Wales captain Bale is on track to win his 100th cap after making progress on his hamstring injury, manager Robert Page said.
Real Madrid's Champions League reign is over.
Real Madrid lift the FIFA trophy for a record fourth time after thrashing local side in Abu Dhabi with the help of impressive distance strikes in either half from midfielders Luka Modric and Marcos Llorente.
Gareth Bale came on as a substitute, but within three minutes of coming on to the pitch in Kiev he had scored the first of his two goals.
The Spanish giants are targeting their fourth Champions League title in five years and their third in a row in the game. Liverpool will be hunting their first trophy in 11 years.
Cristiano Ronaldo converted an injury-time penalty to keep Juventus from pulling off another stunning comeback in the quarterfinals as Madrid advanced despite a 3-1 home loss.
Spanish football club Real Madrid beat arch rivals Barcelona 2-0 to lift the Spanish Super Cup for the 10th time.
A quick preview of key players that are worth watching during this Saturday's UEFA Champions League final.
Gareth Bale and team mates dominate City to secure 1-0 win.
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