"I don't know how long it's going to be,'' Dodgers manager Dave Roberts said of Kershaw's absence. "We're trying to be smart about this and understand that we have the All-Star break. Obviously, there is a lot left of the season, so pushing him doesn't make a whole lot of sense.''
Kershaw is 9-7 with a 3.39 ERA in 106 innings this season. The 33-year-old left-hander is tied for the NL lead in starts with 18.
In his previous start, he allowed three runs in four innings at Washington on July 3, throwing 54 pitches in a rain-shortened game. He had been scheduled to start against Arizona on Friday.
Los Angeles recalled right-hander Mitch White from Triple-A Oklahoma City.
The Associated Press contributed to this report.