A refractive index sensor based on Fano resonances in metal-insulator-metal (MIM) waveguides coupled with rectangular and dual side rings resonators is proposed. The sensing properties are numerically simulated by the finite element method (FEM). For the interaction of the narrow-band spectral response and the broadband spectral response caused by the side-coupled resonators and the rectangular resonator, respectively, the transmission spectra exhibit a sharp and asymmetric profile. Results are analyzed using the coupled-mode theory based on the transmission line theory. The coupled mode theory is employed to explain the Fano resonance effect. The results show that with an increase in the refractive index of the fill dielectric material in the slot of the system, the Fano resonance peak exhibits a remarkable red shift. Through the optimization of structural parameters, we achieve a theoretical value of the refractive index sensitivity (S) as high as 1160 nm/RIU, and the corresponding sensing resolution is 8.62 × 10-5 RIU. In addition, the coupled MIM waveguide structure can be easily extended to other similar compact structures to realize the sensing task and integrated with other photonic devices at the chip scale. This work paves the way toward the sensitive nanometer scale refractive index sensor for design and application.