Direct formation of high-quality and wafer scale graphene thin layers on insulating gate dielectrics such as SiO2 is emergent for graphene electronics using Si-wafer compatible fabrication. Here, we report that in a chemical vapor deposition process the carbon species dissociated on Cu surfaces not only result in graphene layers on top of the catalytic Cu thin films but also diffuse through Cu grain boundaries to the interface between Cu and underlying dielectrics. Optimization of the process parameters leads to a continuous and large-area graphene thin layers directly formed on top of the dielectrics. The bottom-gated transistor characteristics for the graphene films have shown quite comparable carrier mobility compared to the top-layer graphene. The proposed method allows us to achieve wafer-sized graphene on versatile insulating substrates without the need of graphene transfer.