Sources told the newspaper Ready, a former Marine who headed a group called U.S. Border Guard and was running for Pinal County sheriff, killed another man, two women and a young girl.
At least one person survived the carnage, and was being treated at Maricopa Medical Center, the Republic said.
While authorities had not named the other victims, Hugo Maderos of Tampa, Fla., told the newspaper they were his ex-wife Lisa; their daughter, Amber; Amber's boyfriend, whose name wasn't reported; and Amber's 18-month-old daughter, Lilly.
Ready lived at the home with Lisa, who was his girlfriend, Maderos said.
It wasn't known what led to the shootings, city police Sgt. Bill Balafas said.
"This is a domestic situation. We do have a witness that our investigators are interviewing," Balafas said.
Two handguns and a shotgun were recovered from the scene. The Republic said initial reports indicated the killer wore body armor.
KPHO-TV in Phoenix said the incident occurred in a residential area near an elementary and junior high school. The schools were both placed on lockdown until mid-afternoon.