We present a nanoscale refractive index plasmonic sensor based on Fano resonance. We simulate and numerically analyze a novel double T-shaped resonator structure made of conductor–insulator waveguides. Our simulation results show that two Fano resonance peaks can be achieved by the interference between a broadband mode in the straight waveguide and a narrowband mode in the T-shaped resonator. The shifts of Fano resonance peaks by changing the sample refractive index in the resonator structure facilitates the design of a refractive index sensor. To attain a sensor with high sensitivity and figure of merit we employ different geometrical parameters for the resonator structure and analyze the transmission spectra of the sensor. By optimizing the sensor structural parameters we achieve a maximum sensitivity of 523.5 nm/RIU and figure of merit of 2×105 for the sensor made of metal–insulator waveguide. By employing graphene at the core–cladding boundary of the waveguides we attain a high sensitivity of 662.3 nm/RIU and figure of merit of 6.6×105 compared to the literature. We employ samples with refractive indices ranging from 1.0 to 1.05 to analyze the sensor capabilities and employ blood plasma samples to analyze the applications of our sensor structure as a biosensor. Simple fabrication, compactness and high sensitivity are the main advantages of the proposed sensor structure.