Officers called to the home at about 8:30 p.m. Monday found the bodies of Ramon Miranda, 38, and Silvia Miranda, 34, and their children, 12-year-old Ramon Jr. and 10-year-old Rayna, according to the San Bernardino County coroner's office.
The woman and the children had been shot several times and the man was shot once in the head, police spokeswoman Martha Guzman-Hurtado said.
Investigators found a handgun.
"Everything that has been confirmed so far indicates it could be a murder-suicide, but that has not been confirmed," Guzman-Hurtado said. "They are questioning other people before making that conclusion."
However, investigators weren't looking for any suspects, she said.
Police were called by a 16-year-old relative who did not live at the home but went there after being unable to reach the others. It was unclear how long they had been out of contact.
Jonathan Garcia, 22, who lives across the street, said he was watching television at about 8:30 p.m. when he heard yelling.
"I hear, 'Get down, get down!' I recognize that as a cop's voice," he said.
Garcia looked through the window and saw police in SWAT gear with submachine guns.
"So I come outside. They're all rushing at the house. There's a younger kid, probably 16 years old. He's screaming and yelling ... he's saying, 'That's my mom! That's my mom!"
Garcia said a woman outside was crying and kept screaming in Spanish, 'They're all dead, they're all dead. He killed them all!"
Garcia said the family had lived in the home about three years.
The bodies were spread throughout the house, police Sgt. Doug Imhof told KNBC-TV and KABC-TV.
Guzman-Hurtado said records show the last time police visited the home was in June 2010 for an unspecified code-enforcement matter.
"It's a sad day in Fontana," Guzman-Hurtado said. "This is one of those days where we're all supposed to be celebrating with family, making new memories with family as we say goodbye to 2013 and welcome the New Year. It's not a day to be mourning lives lost."
Fontana is a city of about 200,000 people in San Bernardino County about 50 miles east of downtown Los Angeles.