The behavior of strong-field ionization rates of neutral atoms in the vicinity of multiphoton ionization thresholds is analyzed using formal collision theory. Our approach, which accounts nonperturbatively for effects of an intense laser field, shows that the ionization rates have a nearly constant behavior below and above each multiphoton threshold and that between such thresholds there are an apparently finite number of rapid oscillations due to resonances with laser-field-modified Rydberg states. This pattern is typical for any atomic target, as we illustrate specifically for hydrogen and neon atoms. The flat behavior of the ionization yield near multiphoton thresholds gives the appearance of an energy shift of the ionization thresholds, which have been postulated in a number of recent studies concerning diverse aspects of above-threshold ionization and high-harmonic generation of atoms. The flat behaviors of the rates near threshold exhibit only a rather weak dependence on the laser-field intensity. Other aspects of the near-threshold behavior of ionization rates and their dependence on the laser-field parameters are also discussed.