The Jazz said in a statement that Conley will miss Game 1 of the series due to a mild right hamstring strain.
Conley suffered the injury during the first half of the Jazz's series-ending Game 5 victory over the Memphis Grizzlies last week. He was a limited participant in Monday's practice before being ruled out Tuesday morning.
"Hopefully we'll get him back as quickly as possible," Jazz head coach Quin Snyder said. "But those [hamstring injuries] are hard to predict."
The 33-year-old Conley, who was an All-Star selection for the first time this season, missed significant time in the regular season because of hamstring issues. He sat out six games in February and nine games in late April and May due to tightness in his right hamstring.
Conley averaged 17.4 points and 8.6 assists for the top-seeded Jazz in his team's first-round series against the Grizzlies.
Game 1 of the best-of-seven series between the Jazz and Clippers is scheduled for Tuesday at 10 p.m. EDT at Vivint Smart Home Arena in Salt Lake City, Utah.