Gibson, 58, earlier this year revealed he has Parkinson's disease. He has been serving as a broadcast analyst of games for the Detroit Tigers after managing the Arizona Diamondbacks from 2010-14, which included being named National League Manager of the Year in 2011.
Gibson played three seasons for the Dodgers, beginning in 1988, when he won the NL Most Valuable Player Award and led the club into its last World Series. His limp-off winning home run in Game 1 off Dennis Eckersley and the Oakland A's is considered one of the most dramatic and impactful in the history of the game.
Geren managed Oakland from 2007-11, where he worked with current Dodgers general manager Farhan Zaidi.
Among other names the Dodgers are rumored to be considering are internal candidates Gabe Kapler, Ron Roenicke and Tim Wallach, and external candidates Dave Roberts, Bud Black, Darin Erstad and Dave Martinez.
Ken Gurnick is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.