The 59-year-old was the first American to manage a Premier League club, but lasted just 85 days before being sacked by the Swans in December 2016.
They won just two of the former USA and Egypt manager's 11 games in charge.
Los Angeles, who are led by an ownership group of 27 people including Hollywood actor Will Ferrell, will join the MLS for the 2017-18 season.
The team, also co-owned by basketball legend Magic Johnson and former Cardiff City chairman Vincent Tan, will play at the purpose built 22,000-seater Banc of California Stadium.
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Former Swansea City boss Bob Bradley has been named as the first manager of MLS team Los Angeles FC.