This work presents the use of dynamical averaging maps to study perturbed orbits around the Galilean moon Callisto. The study is made in the context of the inner restricted three-body problem, where a small object orbiting the central body, this one assumed to be Callisto, is under the effect of different disturbing forces. Among these forces assumed in the system are: the forces due to the irregular shape of Callisto, which includes its flattening at the poles and its equatorial ellipticity, and the forces due to the gravitational attraction of Jupiter. Regions with different initial orbital distances, inclinations and orbit orientations are mapped and analysed. Results identifying less disturbed regions are obtained using the semi-analytical models adopted. These results are important for the validation and design of trajectories for the missions that are intended to visit the system of Galilean moons in a near future.