The LA Times cited unnamed law enforcement sources saying the driver had told officers he was sleepy, and tests were underway to establish if the 22-year-old was under the influence of any drugs.
Marijuana is legal for recreational purposes in California, and its use is widespread, particularly among younger people.
California Governor Gavin Newsom said he and his wife, Jennifer Siebel Newsom, were thinking of the injured recruits.
"Our hearts are with the Los Angeles County sheriff's recruits injured this morning while training to serve their communities," a statement said.
The group was made up mostly of recruits from the Los Angeles Sheriff's Department, but also included trainees from other nearby police forces.