During the process of synthesizing h-BN on Cu foil via chemical vapor deposition (CVD), low-pressure CVD (LPCVD) typically synthesizes monolayer h-BN films, whereas atmospheric pressure CVD (APCVD) yields few-layer h-BN films. Herein, a growth mechanism for monolayer and few-layer h-BN on Cu foil is proposed using first-principles calculations: Cu(111) passivated h-BN hinders the diffusion of B and N atoms at the subsurface of Cu(111), preventing sufficient transportation of B and N atoms to the existing h-BN layer, thereby leading to the formation of monolayer h-BN in LPCVD. For APCVD, the edges of h-BN are passivated by H, which decreases the barrier energy for the diffusion of B and N atoms on the Cu(111) subsurface, and B and N atoms can easily migrate from the subsurface of Cu(111) to its surface, resulting in the nucleation of h-BN between the existing h-BN and Cu(111), and leading to the formation of few-layer h-BN films. This work provides a theoretical basis at the atomic scale for further understanding the growth of monolayer and few-layer h-BN films on Cu foil.