Based on the Hamiltonian of carriers in bilayer graphene, characteristics of the guided modes in asymmetric quantum-well waveguides were investigated. There exist three different kinds of motions depending on the incident energy of the electrons and external potentials, i.e., quasi-classical motion, quantum Klein tunneling, and mixing of the former two. Given the wave functions for the different regions and using the continuity conditions at interfaces of the potential barriers, the dispersion relationship for the guided modes in asymmetric waveguides was derived analytically. Meanwhile, the structure feature of the guided modes under three different cases was discussed. The result is significant for the practical applications of graphene-based waveguide devices.