A novel nanosensor based on a metal-insulator-metal waveguide coupled to a double ring resonator is proposed. The spectral characteristics are studied by finite element method, and a Fano resonance (FR) formed by the interference of the narrowband mode and the broadband mode is discovered. After analyzing the effects of structural parameters on the transmission characteristics, the structure is further optimized by adding a rectangular cavity inside the ring cavity. The maximum sensitivity reaches 1885 nm/RIU with the figure of merit (FOM) of 77. Additionally, a tunable multiple FR system is realized through the derivative structure, which leads to the splitting of the resonance mode and produces two new narrowband modes. Their formation mechanism and performance are studied through a normalized magnetic field distribution and transmission spectrum. The designed structure with excellent performance can discover significant applications in the future nanosensing domain.