Optical properties of the concentric composite nanostructure composed of gold nanotube around the center gold elliptical core are investigated based on the finite difference time domain method. According to the simulated absorption and scattering spectra, electric field distributions and charge distributions, we can generate double dipole-dipole Fano resonances by adjusting the angle between the elliptical cylinder core and the linearly polarized excitation light, which is due mainly to the interference between the subradiant dipole mode and the superradiant dipole mode. The narrow, low-energy subradiant mode originates from the symmetric hybrization between the longitudinal or transverse dipole mode of the elliptical cylinder core and the dipole bonding mode of the nanotube, and the broad, high-energy superradiant mode originates from the symmetric hybrization between the core’s dipole mode and the nanotube’s dipole antibonding mode. Moreover, the intensities and spectral positions of the two Fano resonances can be manipulated by modifying the geometric parameters of the composite structure. By increasing the semiminor axis of elliptical core, the high-energy Fano resonance red-shifts faster than the low-energy Fano resonance due to the increase of the interaction coupling between the transverse dipole mode of the core and the dipole mode of the nanotube, and becomes weaker in the scattering spectrum because of the reduced radiation intensity of the superradiant dipole mode. When the semimajor axis is changed, a similar phenomenon occurs in the low-energy Fano resonance. In addition, the two Fano resonances red-shift when outer radius of the nanotube increases, but the shift of low-frequency and high-frequency Fano resonance are inconsistent as the inner radius of the nanotube changes. The high-frequency Fano resonance red-shifts monotonically while the low-frequency Fano resonance first blue-shifts and then red-shifts with the increase of inner radius of nanotube because the red shift of the dipole bonding nanotube mode competes with the spectral shifts induced by the diminishing hybridization between elliptical core and nanotube mode. It can also be concluded that the dipole-dipole Fano resonances become apparent and higher order Fano resonance occurs when the composite nanostructure is scaled to a larger size due to the increased radiative damping. With the core and nanotube size fixed, Fano resonance is insensitive to the change of the external environment, but has a good response to the nuclear material of the nanotube.