La Russa, 76, has spoken with general manager Rick Hahn and executive vice president Kenny Williams in recent days.
Levine noted the club hopes to have a new manager in place within the next two weeks.
La Russa retired from managing after winning the World Series with St. Louis in 2011—his second with the Cardinals third overall—and has spent time advising the Arizona Diamondbacks, Boston Red Sox and Los Angeles Angels.
In conversations with White Sox brass, the front office reportedly asked "pointed questions about La Russa's vision and his readiness to handle the White Sox's talented young players", Levine reported, adding the club wants the next manager to provide input on offseason moves.
After announcing Renteria's departure after a 35-25 season that ended with an exit from the American League Wild Card Round, Hahn was very specific when asked what he looks for in a replacement.
"I think the best candidate or the ideal candidate is going to be someone who has experience with a championship organization in recent years," Hahn said. "Recent October experience with a championship organization would be ideal. But we're going to keep an open mind."
The GM mentioned not all candidates would be available to interview right away, as some were still taking part in the postseason. Another oft-mentioned candidate, AJ Hinch, is suspended until the conclusion of the World Series as part of MLB's punishment for the Houston Astros' cheating scandal.
It's unclear if the Sox have met with any other candidates at this point, though Hahn said Chicago is positioned as an ideal opening after reaching the postseason for the first time since 2008 with a young core featuring Tim Anderson, Yoan Moncada, Lucas Giolito and Luis Robert ready to contend.