Kershaw, 30, signed a seven-year, $215 million contract in January 2014 that included a player option to become a free agent after five seasons. The original deadline for Kershaw's decision was midnight ET on Wednesday.
Kershaw, 30, is owed $65 million over the next two seasons.
Kershaw could opt out for the opportunity to stretch out a similar average annual value over additional seasons.
His agreement with the Dodgers to extend his deadline appears to be a sign that both sides want something similar.
The Dodgers also agreed with veteran infielder David Freese to extend the deadline on his option decision until Friday at 4 p.m. ET.
Kershaw, who will turn 31 in March, is probably the greatest pitcher of his generation, and he posted a 2.73 ERA in 2018. But back issues have sent him to the disabled list during each of the past three seasons, and his fastball velocity has declined into the low 90s mph, leading to fewer strikeouts and more balls in play.
"I haven't made the decision yet" Kershaw said after Game 5 of the World Series on Sunday.
The only pitcher who makes more than the $32.5 million Kershaw would average over the next two years under his current deal is former teammate Zack Greinke, who signed a six-year, $206.5 million contract with the Arizona Diamondbacks when he was a year older than Kershaw is now.