We have performed a study of simultaneous nonlinear optical second harmonic generation (SHG) and angle-resolved nonlinear photoemission at the Cu(001) surface excited by fundamental photon energies near 3 eV. At these excitation energies, we identify a dominant contribution to the SHG signal that is determined by a two-photon resonance between the d bands and the n=1 image-potential state at the Cu(001) surface. The near-resonant behavior of SHG and nonlinear photoemission is studied via Cs adsorption, which allows to systematically lower the work function and thus the reference vacuum level of the image potential. Comparison of the angle-resolved photoemission signal, arising from a restricted region of electronic momentum space, with the simultaneous optical SHG signal allows to identify an additional contribution of electronic states with nonzero surface-parallel momentum to the SHG signal enhancements. A simple model accounting for the observed behavior is developed and the implications for the quantitative understanding of SHG are discussed.