In Davis-Januszkiewica´s paper Hyperbolization of polyhedra it is shown that for every manifold $M$ there exists a map $N \to M$ of non-zero degree such that $N$ is aspherical (plus some more properties of such a map). They also say that such a manifold $N$ has "non-positive" curvature.

My question is whether one can chose $N$ to be negatively curved, or at least that $N$ has word-hyperbolic fundamental group. Or, if the requirement that $N$ is aspherical is too strong, whether every manifold is dominated by a manifold with word-hyperbolic fundamental group.