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.
