This paper initiates the study of the problem of constructing in-block controllable covers of nonlinear systems on polytopes, which is useful in studying approximate mutual accessibility problems of nonlinear systems on polytopes. In particular, for a given nonlinear system and a given polytope, we study how to systematically construct polytopic covers of the given polytope such that the affine systems, resulting from the linearization of the nonlinear system inside the polytopic regions of the cover, are in-block controllable. By exploring the geometry of the problem, we first provide some constructive guidelines for building such covers. Then, for a special class of nonlinear systems, we provide a constructive algorithm. We also give illustrative examples of the main results.