The effective interaction hyperspherical-harmonics method, formulated for local forces, is generalized to accommodate nonlocal interactions. As for local potentials this formulation retains the separation of the hyper-radial part leading solely to a hyperspherical effective interaction. By applying the method to study ground-state properties of He4 with a modern effective-field-theory nucleon-nucleon potential model (Idaho-N3LO), one finds a substantial acceleration in the convergence rate of the hyperspherical-harmonics series. Also studied are the binding energies of the six-body nuclei He6 and Li6 with the JISP16 nuclear force. Again an excellent convergence is observed.