Following the framework of the variable-phase approach, we derive an equation for determining the scattering amplitude of a non-relativistic quantum particle in a non-local potential. Its solution implies the integration of the Volterra integro-differential equation of the first kind and allows determination of bound-state energies and wavefunctions. A fast numerical scheme for the solution of these equations is suggested and it is demonstrated that the proposed method requires the numerical efforts of the same order as in the local potential case.