Compared to metallic nanostructures employed in plasmonics, dielectric materials with high refractive index can directly engineer magnetic responses in addition to the electric responses in plasmonic structures. In this work, we have demonstrated that the magnetic-based double Fano resonances can be achieved in the Au-SiO2-Si multilayer nanoshells due to the strong interaction between electric resonances and magnetic resonances. The observed double Fano resonances arise from two distinct origins: (1) the interference of magnetic dipole resonance of Si core and electric dipole resonance of Au shell, which excites a Fano resonance even in a symmetric nanoshell, and (2) the interference between magnetic quadrupole resonance of Si core and electric quadrupole resonance of Au shell, introduced by symmetry breaking of Si core. Dipole radiative enhancement spectra are used to analyze electric and magnetic responses of Au-SiO2-Si multilayer nanoshells, and the asymmetric Fano line shape is fitted by a coupled oscillator model. Also, magnetic-based Fano resonances can be adjusted easily by means of varying the geometrical parameters. The spectral sensitivity of Au-SiO2-Si multilayer nanoshells is also investigated, and the Fano resonance modulated from scratch can be found when the intermediate dielectric layer between Au shell and Si core is altered. The Au-SiO2-Si multilayer nanoshells that show better tunability of magnetic-based Fano resonances may provide the various applications ranging from novel optical devices to biological sensors.