Thermal transport in the Si/SiO₂ multishell nanotubes is investigated theoretically. The phonon energy spectra are obtained using the atomistic lattice dynamics approach. Thermal conductivity is calculated using the Boltzmann transport equation within the relaxation time approximation. Redistribution of the vibrational spectra in multishell nanotubes leads to a decrease of the phonon group velocity and the thermal conductivity as compared to homogeneous Si nanowires. Phonon scattering on the Si/SiO₂ interfaces is another key factor of strong reduction of the thermal conductivity in these structures (down to 0.2 Wm^-1K^-1 at room temperature). We demonstrate that phonon thermal transport in Si/SiO₂ nanotubes can be efficiently suppressed by a proper choice of nanotube geometrical parameters: lateral cross section, thickness and number of shells. We argue that such nanotubes have prospective applications in modern electronics, in cases when low heat conduction is required.