Epsilon-near-zero (ENZ) metamaterial with the relative permittivity approaching zero has been a hot research topic for decades. The wave in the ENZ region has infinite phase velocity (v=1/ε μ), but it cannot efficiently travel into the other devices or air due to the impedance mismatch or near-zero group velocity. In this paper, we demonstrate that the tunable index-near-zero (INZ) modes with vanishing wavenumbers (k = 0) and nonzero group velocities (vg ≠ ~0) can be achieved in nonreciprocal magneto-optical systems. The INZ modes have been experimentally demonstrated in the photonic crystals at Dirac point frequencies, and that impedance-matching effect has been observed as well [Nat. Commun.8, 14871 (2017)10.1038/ncomms14871]. Our theoretical analysis reveals that the INZ modes exhibit tunability when changing the parameters of the one-way (nonreciprocal) waveguides. Moreover, owing to the zero-phase-shift characteristic and decreasing vg of the INZ modes, several perfect optical buffers are proposed in the microwave and terahertz regimes. The theoretical results are further verified by the numerical simulations using the finite element method. Our findings may open new avenues for research in the areas of ultra-strong or -fast nonlinearity, perfect cloaking, high-resolution holographic imaging, and wireless communications.