Jones, the reigning UFC light heavyweight champion, is charged with leaving the scene of an accident involving great bodily harm, Albuquerque police said.
A pregnant woman in her 20s, who was driving one of the other vehicles, was injured in the accident, police said.
He was booked into the Bernalillo County Metropolitan Detention Center on Monday.
According to the police report, a silver SUV made an illegal turn at a red light, hitting a passing car and knocking that car into a third vehicle.
Witnesses reported seeing a man get out of the SUV and flee, only to return and grab "a large hand full of cash" from the SUV. He then shoved the cash into his pants before jumping a fence and running away, the police report states.
Officers searched the SUV and found car rental forms with the name Jonathan Jones, as well as MMA infomation on them.
The police report also says a marijuana pipe with marijuana inside it was found in the vehicle, and the report lists "drugs or medication" among apparent factors contributing to the accident.
Ultimate Fighting Championship issued a statement saying it was gathering facts and reserving comment until more information is available.
CNN Sports has reached out to Jones' attorney for comment.
Jones is arguably the UFC's biggest star in the sport's most respected weight class.
But he has also had his struggles. In January, Jones entered a drug rehabilitation center after testing positive for cocaine in a test administered by the Nevada Athletic Commission. He checked himself out of rehab after one day of treatment.
Jones is scheduled to fight Anthony Johnson in UFC 187
on May 23 in Las Vegas.