Bloodstains found in the family's house near the city of Nantes match the DNA of three of them, officials say.
There has been no activity in any of their bank accounts or mobile phones since they went missing, they add.
Reports suggest the father once suffered from depression while his son had psychological problems.
The Troadec family - Pascal and Brigitte, both aged around 50, their son Sebastien, 21, and daughter Charlotte, 18 - were last seen on 16 February. Authorities were informed of the family's disappearance by Brigitte's sister.
Investigators searched their two-storey house and found bloodstains on Sebastien's phone and on Brigitte's watch, prosecutor Pierre Sennes said.
Traces of blood were also found under the stairs and, according to Mr Sennes, it appeared that someone had tried to wipe them away.
The beds had been stripped, with some sheets drying on an indoor rack. Wet clothes were still in the washing machine and food was going off in the refrigerator.
"It's as if the life of the house was frozen in time," Mr Sennes said.
The family's two cars were still parked in front of the house but investigators have not yet found Sebastien's car.
Sebastien had been sentenced to carry out community service after being convicted in 2013, when he was a minor, for making death threats on his blog, Mr Sennes added.
French newspaper Le Monde reported that Sebastien had posted on a video game online forum: "What do I hate the most? My reflection." And: "What makes me sad? My life."
And on Twitter, he had written: "If one really knew what was going on in my head, one would think I was a crazy person with no morals."
An investigation into murder, abduction and illegal confinement has been opened, the prosecutor says.