The Times said Vela - the reigning MLS scoring champion and Most Valuable Player - had opted to skip LAFC's trip to Florida to remain at home with his pregnant wife and young son.
Major League Soccer allows players to withdraw from the "MLS is Back" tournament for health or family reasons.
Mexico international Vela's withdrawal robs the league's restart tournament of one of the league's biggest names just two days before the competition is set to kick off in Orlando.
All 26 MLS teams are taking part in the tournament, which will start with a group stage followed by a knockout competition.
Teams are being based inside a protective "bubble" at the Walt Disney World Resort sports complex in Orlando for the duration of the tournament, which concludes on 11 August.