The person, confirming a USA Today report, spoke on the condition of anonymity because the Angels typically do not comment on personnel matters.
A three-time World Series-winning manager, the 76-year-old La Russa joined the Angels prior to this season as senior advisor of baseball operations.
The White Sox agreed to split with Rick Renteria after a disappointing finish to a breakthrough season in which Chicago made the playoffs for the first time since 2008.
La Russa, who started his managing career with the White Sox during the 1979 season, hasn't managed since 2011, when he led St. Louis past Texas in the World Series. He also won championships with Oakland in 1989 and the Cardinals in 2006.
If hired, he would be the oldest manager in the major leagues by five years. Houston Astros manager Dusty Baker is 71.
La Russa is 2,728-2,365 with six pennants over 33 seasons with Chicago, Oakland and St. Louis and was enshrined in Cooperstown in 2014. Only Hall of Famers Connie Mack (3,731) and John McGraw (2,763) have more victories.
LaRussa managed the White Sox to a 522-510 record over parts of eight seasons. He led the 1983 team to 99 wins and the AL West championship. But he was fired in 1986 by then-general manager Ken Harrelson after the White Sox got off to a 26-38 start.
Chairman Jerry Reinsdorf has long regretted that move and remains close with La Russa. If La Russa gets the job, he will inherit a team that appears poised for long-term success.
The White Sox have never made back-to-back playoff appearances. But after ending a string of seven losing seasons, they are in position to change that.
They have a core of young players on team-friendly deals, starting with shortstop Tim Anderson. Veteran Jose Abreu put himself in the running for AL MVP by driving in 60 runs. Ace Lucas Giolito pitched his first no-hitter.
Eloy Jimenez hit .296 with 14 homers and 41 RBIs. Luis Robert, who agreed to a $50 million, six-year contract in January, showed star potential in a roller-coaster rookie year. He got off to a great start and hit a massive homer in the playoff series against Oakland. But he also slumped in September.