The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points <i>F</i>(<i>S</i>) of a nonexpansive mapping <i>S</i> and the set of solutions <sub> <i>A</i> </sub> of the variational inequality for a monotone, Lipschitz continuous mapping <i>A</i>. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of {F(S)\cap\Omega_{A}}. As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.