Graphene-based materials with a negative Poisson's ratio have numerous potential applications in various fields. However, the modification of graphene is cumbersome and may worsen the mechanical properties. The scale limitation and structural instabilities of suspended graphene are also unfavorable for practical applications. In the present study, we design several nanolayered graphene/Cu composites and investigate their tensile behavior using molecular dynamics simulations. The nanolayered composites exhibit an apparent auxetic behavior without any modification of graphene, as the graphene/Cu interface can significantly enhance the surface effect and lead to an earlier phase transformation of the Cu component. A simultaneous occurrence of a positive and negative Poisson's ratio can be achieved in an asymmetric composite due to the good blocking effect of graphene on two separated Cu films. Materials with simultaneous negative/positive Poisson's ratio have potential applications in scaffold design, where it is necessary to tune the magnitude and polarity of the Poisson's ratio in tissue engineering. Furthermore, we propose a composite consisting of alternating multilayer graphene and thin Cu films to overcome the scale limitation, whose negative Poisson's ratio persists when the total thickness exceeds 100 nm. It is found that the change in the absolute value of Poisson's ratio becomes smaller with an increase in total thickness. Graphene/Cu composites with only a slight deformation under external loading may be suitable for the fabrication of telecommunication cables, whose dimensions should remain unchanged when subjected to high hydrostatic pressure in the deep ocean.