Ken Gurnick of MLB.com reported the news, noting Zaidi wouldn't specifically mention whether the two sides have discussed Kershaw's contract but that Zaidi still called Kershaw "our franchise player" when mentioning the dialogue.
The southpaw has an out clause in his contract and could test free agency following the 2018 season.
Kershaw will turn 30 on March 19 but showed few signs of slowing in 2017, notching a 2.31 ERA, 0.95 WHIP and 202 strikeouts in 175 innings. The last time he had an ERA above 2.91 was during his rookie year in 2008, and he has posted an ERA below 2.00 three times.
He already has a Hall of Fame resume as the 2014 National League MVP, a three-time NL Cy Young Award winner, a five-time NL ERA titlist and a seven-time All-Star and likely has plenty of productive years remaining given his recent outputs, though he has missed time with back injuries the last two seasons.
Kershaw has been with the Dodgers his entire career, and it's difficult to envision him wearing any other uniform. He is arguably the most dominant force in the sport, however, and would almost assuredly generate massive interest and offers on the open market.
His immediate future figures to be one of the most intriguing storylines of the offseason if he doesn't work something out with Los Angeles, as his presence alone could dramatically reshape any starting rotation.