The computation offloading problem of multi-access edge computing (MEC) has become one of the research focuses. The current computation offloading scheme only considers the computation offloading problem in the cloud, edge, and end structures and does not take into account the attributes of the public and private clouds. In this study, a novel computation offloading scheme is proposed, which considers the relationship between the public cloud and private cloud in edge computing and regards the public cloud as a supplement to private cloud resources to alleviate the insufficient computing power caused by the limitations of private cloud resources. Moreover, a two-layer Stackelberg game is established to solve the computation offloading problem. The optimal strategies of each player are obtained upon the analysis of the strategies and profits of the public cloud, the private cloud, and users, and the existence and uniqueness of the Nash equilibrium solution to the two-layer game are proved. The simulation results and analysis verifies the feasibility of the computation offloading scheme based on the two-layer Stackelberg game. Compared with the computation offloading scheme based on the single-layer Stackelberg game, the proposed scheme is more efficient and more suitable for edge computing environments.