The team said Friday it would provide updates as appropriate.
Leonard suffered an apparent head injury on Thursday night, late in the third quarter of the Spurs' 102-92 loss to the Oklahoma City Thunder.
Spurs coach Gregg Popovich said after the game that he wasn't sure what had happened to his All-Star forward.
"He got hit in the mouth or the cheek or the head or something. I'm not sure which. So the trainer thought he should sit," Popovich said.
Leonard was struck on the left side of his head by Victor Oladipo as the Thunder guard flailed his arms after losing the ball on a drive to the basket.
Leonard knocked down a pair of free throws at the 2:29 mark of the third quarter, before walking off the court to the locker room with a trainer only to never return to the floor. The club originally anticipated that Leonard would not go into the NBA's concussion protocol but changed course after performing additional evaluations.
Leonard's injury came during a contest that a few Spurs described as "chippy" due to the physicality between the teams throughout.
He cannot return until he is evaluated by a team physician and passes the NBA's concussion protocol. Players must be symptom-free as they undergo a series of tests that include riding a stationary bike, jogging, agility work and noncontact team drills.
The loss of Leonard comes at an inopportune time, as the Spurs hold the No. 2 seed in the Western Conference and sit two games behind Golden State with a chance to gain ground when the teams meet on Saturday at the AT&T Center with the Warriors coming off a Friday night matchup against the Minnesota Timberwolves. A win over the Warriors on Saturday would give the Spurs the season series between the teams, as they defeated Golden State 129-100 in the season opener. Clinching the season series would prove beneficial for the Spurs should the teams finish the regular season in a tie for the No. 1 seed.
"Next man up," starting shooting guard Danny Green said. "It's tough. He's a big percentage of our offense and defense. Just like last game [against the Sacramento Kings on Wednesday], we had a game with no Kawhi and no [LaMarcus Aldridge]. You just have to find a way."
That becomes more difficult without Leonard in the fold.
Without Leonard on the floor in the first half Thursday at Oklahoma City, San Antonio managed a minus-8 points differential and shot 27 percent while putting together an assist-to-turnover ratio of 1-to-3. With Leonard on the floor, the Spurs' points differential was plus-2, they shot 53 percent and they produced an assist-to-turnover ratio of 9-to-4.
The Spurs could face the Warriors without both Leonard and point guard Tony Parker, who is listed as questionable due to back stiffness. Parker was held out of Thursday's game against the Thunder.
San Antonio now must lean on Aldridge to create offense, along with veteran center Pau Gasol. Aldridge is the Spurs' second-leading scorer, averaging 17.3 points per game, while Gasol averages 12.1.
"Oh, we don't get to sit that one out?" veteran David Lee jokingly asked. "We're going to have a fun one on Saturday. It's going to be a battle just like this one was. We want to win every game. We have to regroup and do better the things we didn't do well enough [against Oklahoma City] and prepare for a good Golden State team coming in."
Leonard, 25, a two-time Defensive Player of the Year, finished Thursday night with 19 points in 26 minutes. He's averaging career highs this season with 26.3 points and 3.4 assists to go along with 6.0 rebounds.
Information from The Associated Press was used in this report.