||We investigate possible topological superconductivity in the Kondo-Kitaev model on the honeycomb lattice, where the Kitaev spin liquid is coupled to conduction electrons via the Kondo coupling. We use the self-consistent Abrikosov-fermion mean-field theory to map out the phase diagram. Upon increasing the Kondo coupling, a first-order transition occurs from the decoupled phase of spin liquid and conduction electrons to a ferromagnetic topological superconductor of Class D with a single chiral Majorana edge mode. This is followed by a second-order transition into a paramagnetic topological superconductor of Class DIII with a single pair of helical Majorana edge modes. These findings offer a route to topological superconductivity in the Kondo lattice system. We discuss the connection between topological nature of the Kitaev spin liquid and topological superconductors obtained in this model.