The suspect, identified as One Goh, surrendered to police who tracked him down in a neighboring city after he fled the scene of the massacre in Oakland, near San Francisco, said Oakland police chief Howard Jordan.

"The suspect, who is believed to be an Oakland resident, surrendered to officers of the Alameda Police Department in their city," he told a press conference, giving the first official details of the massacre.

"He is a Korean national," he added, without specifying whether the suspect was from South or North Korea.

The alleged shooter had apparently fled the scene before officers, responding to a series of 911 calls around 10:30 am, arrived at the Oikos University in East Oakland to scenes of chaos.

A total of 10 people were hit, five of whom were pronounced dead at the scene. Two others died later in hospital. Other people were found hiding in classrooms at the small Christian college, he said.

The gunman was arrested about five miles away near a Safeway supermarket in Alameda County, according to local media. CNN reported that he told the store's customer services counter that he had just shot a number of people.

"He actually commandeered a victim's car, and drove it to Alameda, and that's where he surrendered," added Jordan, describing the college as a "very bloody scene."