"Jones tossed his helmet to side in frustration after he was hurt," reported Eric Branch of the San Francisco Chronicle.
Rookie defensive tackle Javon Kinlaw replaced Jones for the remainder of practice.
Cornerback K'Waun Williams suffered a Grade 1 calf strain over the weekend and is considered week-to-week.
"I don't expect to have him here, probably for about two weeks," Shanahan said.
Shanahan said three players — Richie James Jr. (hand), center Weston Richburg (patellar tendon), and defensive lineman Jullian Taylor (ACL) — aren't expected to be healthy by the start of the season. James could start the season on the non-football-injury (NFI) list while Richburg and Taylor could end up on the physically-unable-to-perform (PUP) list.
General manager John Lynch had already confirmed that Richburg would likely land on PUP.
Shanahan remains hopeful that his second-year wide receiver, Deebo Samuel (foot), could be healthy enough to play during Week 1's matchup at Levi's Stadium against the Arizona Cardinals. That optimism extended to defensive lineman Ronald Blair (ACL).