The interview took place on Monday, according to the sources.
The Dodgers have been looking for a replacement since Don Mattingly and the club mutually agreed to part ways last month.
Gibson spent three years as a player with the Dodgers. His walk-off home run in Game 1 of 1988 World Series helped Los Angeles to its last championship, and still rates as one of baseball's great moments.
Gibson managed the Arizona Diamondbacks from 2010 to '14, and was named the NL's top manager in 2011 after leading the team to 94 wins. He was fired in September 2014, just days before the Diamondbacks finished a 64-98 season.
This past April, Gibson revealed that he has been diagnosed with Parkinson's disease.
"I have faced many different obstacles in my life, and have always maintained a strong belief that no matter the circumstances, I could overcome those obstacles," Gibson said in a statement then. "While this diagnosis poses a new kind of challenge for me, I intend to stay true to my beliefs.
"With the support of my family and friends, I will meet this challenge with the same determination and unwavering intensity that I have displayed in all of my endeavors in life. I look forward to being back at the ballpark as soon as possible."
Gibson, 58, had been calling games for the Detroit Tigers for Fox Sports Detroit.
He could not be reached for comment regarding the Dodgers' position.
As many as 10 people are expected to interview for the job, sources told ESPN's Mark Saxon.
Former San Diego Padres bench coach Dave Roberts already has interviewed, and University of Nebraska coach Darin Erstad reportedly will interview this week, sources said. Gabe Kapler, Ron Roenicke and Tim Wallach are expected to be candidates from in-house, sources said.
Chicago Cubs bench coach Dave Martinez and New York Mets bench coach Bob Geren also are expected to receive interviews. On Wednesday, Mets general manager Sandy Alderson acknowledged giving permission to Geren to speak with the team.
Bud Black, who was in the running for the managerial job with the Washington Nationals before Dusty Baker was hired, also reportedly will meet with the Dodgers.
Kirk Gibson a 'perfect fit' for the Dodgers
ESPN baseball Insider Jim Bowden explains why he believes Kirk Gibson would be a great manager for the Dodgers.