In this paper, we report a numerical investigation of tunable dual-band metamaterial perfect absorber (MMPA) consisting of two-intersecting graphene nanorings arrays. The optical absorption performance of absorber is dominated by the inner/outer radius of nanorings and center distance between the two nanorings. For the resonance modes λ1 and λ2, the sensitivity of probing the surrounding refractive index can reach 7.97 μm/RIU and 13.27 μm/RIU respectively. The absorption performance of the under-study graphene nanorings can be flexibly adjusted by modulating the Fermi level of graphene with voltage. Due to excellent wavelength selectivity and high refractive-index sensitivity, the presented tunable dual-band MMPA has great application prospects as sensors in biopharmaceutical, environmental monitoring.