The 33-year-old will reportedly cost a Major League Soccer record $26 million (£18.7m) transfer fee, according to ESPN and The Athletic.
"I'm incredibly proud to be joining LAFC, a club with big ambitions in one of the most iconic sports cities in the world," he said.
"Los Angeles has such a rich history of champions, and I am here to help write the next chapter. I'm excited for this new challenge in MLS. I have come to LA to lift trophies and give everything for this club, this city, and its fans."
He will reunite with former teammate Hugo Lloris at the club, who are currently sixth in the Western Conference having played 22 matches. They're 10 points off leaders San Diego with three games in hand.
'One of the greatest'
The South Korean forward departs England as one of Tottenham's greatest ever players after 10 years, 454 games and 173 goals.
He had the dream ending to his time at Spurs, captaining the club to Europa League glory at the end of last season, ending their 17-year wait for a trophy.
"Sonny is one of the greatest players to ever wear the famous Lilywhite shirt and he has been a joy to watch over the past decade," said chairman Daniel Levy.
"Not only is he a wonderfully gifted footballer, but he's also an incredible human being who has touched hearts and inspired people throughout the club and all over the world.
"Sonny has given so much to this club, both on and off the pitch, and for that we are forever grateful.
"We wish him all the best for the future, and he will always be welcome back at the club as a beloved and cherished member of our Spurs family."