The two-dimensional material graphene is usually required to be transferred on the target substrate for some special applications, thus it is important to understand the adsorption properties in the graphene transferring and stripping processes. In this paper, the adsorption properties of a single-layered graphene on the grooved copper substrate are investigated using molecular dynamics simulations. The influence of geometric characteristic size of the groove on the adsorption force of the graphene deriving from the substrate is explored. For the fixed boundary conditions of the graphene, the adsorption force increases up to maximum and then decreases with reducing the distance between the graphene and substrate in the adsorbing process. The maximum adsorption force increases with groove depth increasing, with the groove width kept constant. Nevertheless, as the groove depth increases continuously, the adsorption force decreases greatly until the graphene cannot be adsorbed into the groove. In the graphene stripping process, the critical force that can strip the graphene completely from the substrate increases first and then decreases with the increase of the groove depth, which is also dependent on the steady adsorbing configuration of the system before stripping. With the groove depth kept constant, the magnitude of the adsorption force between the graphene and substrate is determined by the steady adsorbing configuration of the graphene in the groove region. The adsorption force versus the distance between the graphene and the grooved substrate can be divided into two groups according to whether the graphene can be adsorbed into the groove. In both adsorbing and stripping processes, the adsorption force for the graphene adsorbed into the groove is obviously larger than that for the graphene covered on the groove. Moreover, the influence of the boundary condition of the graphene on the adsorption properties in the groove region on the substrate is considered preliminarily. It indicates that the tensile plane stress within the graphene sheet induced by the fixed boundaries can hinder the graphene from being adsorbed into the groove. The findings may be helpful for the graphene-based fabrication of nano-apparatus and functionalized surface modification.