The accessibility properties of time-variant nonlinear control systems with fast time variables are investigated. It turns out that local accessibility can be carried from a time-invariant averaged system to the original system, provided the time variable is sufficiently fast, or equivalently, the singular perturbation parameter is sufficiently small. It is shown that the reachable sets of the averaged and the original systems contain common open sets. To this end, the singular perturbation parameter is considered as a homotopy parameter, and mapping degree methods are used. An application to controllability properties of time-variant nonlinear control systems with fast time variables is presented as well.