Here we study the approximate controllability for control problems driven by a class of nonlinear evolution hemivariational inequalities in Hilbert spaces. Actually, our results cover a broader class of inclusion problems involving time-dependent operators. First, by using a fixed point approach and nonsmooth analysis, we show the existence of mild solutions for the inequality problem. Next, we provide a sufficient condition guaranteeing the approximate controllability of our problem in terms of an associated linear control system. An example demonstrates the applicability of our results.