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The official website of MLS claims the winger will join LAFC on a one-year contract using targeted allocation money.
That deal would expire next June, midway through the 2023 season.
Bale's Real Madrid contract expires this month, and he has been looking for a new club after guiding Wales to qualification for the 2022 FIFA World Cup.
The five-time UEFA Champions League winner has been widely linked with Cardiff City but instead appears set on a move Stateside.
Wales faces United States in its first group-stage match at the FIFA World Cup in Qatar later this year.
Before then, Bale's signing would boost an LAFC team which already leads the Supporters' Shield race and has also secured Italy defender Giorgio Chiellini ahead of the transfer window in MLS opening next month.
LAFC's first match after that date is El Trafico against rival the LA Galaxy.
