Graphene growth on Cu foil was demonstrated by plasma enhanced chemical vapor deposition (PECVD) with a microwave-excited non-equilibrium atmospheric pressure remote plasma (MNAPP) using CH4 and H2 gases diluted by He gas. The density of graphene grain decreased and their average sizes increased with decreasing the CH4 flow rate. Moreover, the sizes were almost constant at each condition. In the MNAPP, emission intensities of CH and C2 radicals normalized by that of He atom decreased with decreasing the CH4 flow rate. From the results, nucleation of graphene on Cu surface caused by the precursors such as CH x and C2 radicals occur only at the early period of PECVD, and after the initial nucleation, these precursors are mainly consumed for the planar growth of graphene without inducing subsequent nucleation. Therefore, flux control of precursors at the early period is important for synthesizing graphene with more large size in PECVD using MNAPP.