In this paper, we indicate that the Sakurai-Sugiura method with Rayleigh-Ritz projection technique, a numerical method for generalized eigenvalue problems, can be extended to nonlinear eigenvalue problems. The target equation is T (λ)v = 0, where T is a matrix-valued function. The method can extract only the eigenvalues within a Jordan curve Γ by converting the original problem to a problem with a smaller dimension. Theoretical validation of the method is discussed, and we describe its application using numerical examples.