Santa Clara police confirmed that McDonald was arrested at 7 a.m. local time Monday in San Jose, where the incident occurred. He was arrested on a charge of domestic violence and a separate charge of child endangerment.
The Bears have not immediately commented on the arrest, but team chairman George McCaskey initially had balked at the notion of signing McDonald earlier this offseason due to past allegations of domestic assault when he played for the 49ers.
McCaskey ultimately allowed general manager Ryan Pace to sign McDonald after the veteran defensive lineman flew to Chicago to meet with Bears management.
Chicago signed the 30-year-old McDonald to a one-year, $1.05 million contract with no guaranteed money in the deal.
San Francisco released McDonald in December because of what 49ers general manager Trent Balke called a "pattern of poor behavior."
Police arrested McDonald last August for suspicion of domestic violence. The Santa Clara County district attorney's office later declined to file charges, however, following a lengthy investigation. The NFL also announced that McDonald would not be subject to league punishment for the August incident.
In December, the San Jose Police Department began investigating McDonald for suspicion of sexual assault. That case remains open.
ESPN.com Bears reporter Jeff Dickerson contributed to this report.