Roberts said Kershaw will not travel with the Dodgers on Friday for this weekend's series in Colorado and will instead stay behind for tests. Roberts said the tightness was in Kershaw's "mid to lower" back.
"With Clayton's history, there is some concern," Roberts said. "We want to make sure that he is well and good."
Kershaw, who had been out since May 1 with left biceps tendinitis, threw 62 pitches over five innings in a 2-1 loss to the Philadelphia Phillies.
He gave up four hits and a run while walking one and striking out five.
"It's beyond frustrating," Kershaw said afterward. "You feel like you're out of the woods and feeling good and then this creeps up on you. I'm frustrated that I can't contribute to the team."
The Dodgers are 2-6 in Kershaw's eight starts this season; they went 23-4 in his starts in 2017.
All 20 of Kershaw's four-seam fastballs were 90.0 mph or slower in his outing Thursday, according to ESPN Stats & Information. Last year, not a single one of his 1,142 four-seam fastballs was 90.0 mph or slower.
Roberts said the tightness in Kershaw's back "could have contributed to the lack of velocity."
Kershaw is 1-4 this season with a 2.76 ERA in 49 innings, with 11 walks and 53 strikeouts.
He has missed significant time in each of the past two seasons -- and three of the past four -- with back injuries. Kershaw said he doesn't believe this flare-up is as bad as the one that landed him on the disabled list for 75 days in 2016 and the one that sidelined him 39 days last season.
"If you compare this to anything, compare it to last year," Kershaw said. "The year before in 2016 was pretty severe."
A three-time NL Cy Young Award winner, Kershaw was sidelined in 2014 from March 23 to May 5 with a major muscle strain in his back; in 2016 from June 27 to Sept. 9 with a herniated disk in his lower back; and in 2017 from July 24 to Sept. 1 with a lower back strain.
Kershaw is earning $30.4 million this year in the fifth season of a $215 million, seven-year contract but has the right to opt out after the season and become a free agent, giving up salaries of $32 million next year and $33 million in 2020.
ESPN's Pedro Gomez and The Associated Press contributed to this report.