In the present paper we consider local center-unstable manifolds at a stationary point for a class of functional differential equations of the form ẋ(t) = f (xt) under assumptions that are designed for application to differential equations with state-dependent delay. Here, we show an attraction property of these manifolds. More precisely, we prove that, after fixing some local center-unstable manifold Wcu of ẋ(t) = f (xt) at some stationary point j, each solution of ẋ(t) = f (xt) which exists and remains sufficiently close to j for all t ³ 0 and which does not belong to Wcu converges exponentially for t ® ¥ to a solution on Wcu