Kershaw now has until 16:00 ET on Friday to decide what he is going to do, the team announced on Wednesday.
The 30-year-old starting pitcher has two years remaining on the seven-year, $215million deal he signed with Los Angeles in 2014.
The Texas native went 9-5 with a 2.73 ERA in 161.1 innings pitched in 2018.
Kershaw has not eclipsed the 200-inning mark since 2015.
He has made seven All-Star games and won the National League Cy Young Award three times.
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