Willow, 13, caused a media frenzy earlier this month by posing in bed with 20-year-old Arias, who was shirtless in the photo. Willow was fully clothed, and Arias is a family friend who's best friends with Willow's older brother, Jaden.
Parents Will Smith and Jada Pinkett Smith denied that there was anything inappropriate about the photo, and said Willow is mature enough to decide who she hangs out with.
"Here's the deal," Pinkett Smith told TMZ on May 7. "There was nothing sexual about that picture or that situation. You guys are projecting your trash onto it. You're acting like covert pedophiles and that's not cool."
"The investigation was formally opened last week and is being taken very seriously by the department," a source told Radar.
"Will and Jada Pinkett Smith have been extremely cooperative with officials. Of course, they aren't happy that their parenting skills are under scrutiny, but they understand."
The same "insider" explained that the inquiry would involve multiple conversations with Willow, and at least one conversation with Arias, best known for his role on Hannah Montana.
"This won't just be one or two visits with the family, and it will likely be an open investigation for at least a month, out of an abundance of caution," the source said.
The department might want to spend less time investigating an Instagram photo with a family friend and more time looking into Willow's experiments with Orgone energy and reading of Indian "sex guru" Osho.