Among the other candidates LAFC expects to interview are Las Vegas Lights coach Steve Cherundolo and LAFC assistant Ante Razov, a source told ESPN.
The job has also generated significant interest abroad and several more candidates are expected to receive consideration.
Osorio managed Mexico from 2015-2018, guiding the team to the 2018 FIFA World Cup, where El Tri advanced out of the group stage, including a win against Germany, before losing 2-0 to Brazil in the Round of 16.
Osorio coached LAFC general manager John Thorrington in 2007, during a brief spell with the Chicago Fire, and coached the New York Red Bulls in 2008 and 2009.
Cherundolo, the former United States international, spent the 2021 season coaching Las Vegas in the USL Championship. The club has a partnership with LAFC that sees the teams train together in Los Angeles. Razov has been with LAFC as assistant since the club's debut season in 2018.
Vela's contract keeps him under club control until the summer transfer window, according to a league source.
It's possible he could move in the January window, extend his contract with LAFC under more team-friendly terms or play through the end of his deal.