Here, we demonstrate a simple method of growing graphene directly on various dielectric substrate using vapor-phase metal catalyst via mobile hot-wire (MHW) assisted chemical vapor deposition (CVD). The MHW made of nickel (Ni) is utilized as an independent source of the metal vapor as well as a moving heat source. The hot-wire temperature (Tw, 800–1100 °C) and the total chamber pressure (Ptot, 0.1–760 torr) determine the equilibrium partial vapor pressure of nickel (PNi, ∼10−11 to ∼ 10−3 torr). When the equilibrium PNi is built in the chamber, reaction between Ni vapor and carbon feedstock in a gaseous phase results in deposition of carbon containing Ni particles on dielectric substrate. The optimum growth conditions for low-defect and uniform graphene are found at the substrate temperature (Tsub) of 700 °C and the speed of MHW (Vw) near 1.0 mm/min, which determines the nucleation and lateral growth of graphene from the deposited Ni particles. Consequently, the PNi was clarified as a primary factor for graphene grown on non-catalytic substrate (NCS) by comparing the graphene grown by solid and vapor-phase metal catalyst. We believe the results contribute to the understanding of the direct-growth mechanism of graphene on NCS.