Hybrid density functional theory has been applied for investigations of the electronic and atomic structure of nanotubes based on titanium and zirconium disulfides. The full optimization of all atomic positions in the considered systems has been performed to study the atomic relaxation and to determine the most favorable nanostructures. Our calculations on single-wall TiS2 and ZrS2 nanotubes confirmed that the nanotubes obtained by rolling of the hexagonal layers of 1T crystalline polymorphs with the octahedral morphology (and with the layer group Pm1) are the most stable. However, it is also possible to obtain the relatively stable nanotubes with lepidocrocite morphology by rolling up layers of the metastable tetragonal or orthorhombic phases. The strain energy of TiS2 is almost the same as that of ZrS2 but it is greater than the strain energy of TiO2 and ZrO2 nanotubes. However, the formation energy of the sulfide nanotubes is considerably less than the formation energy of the oxide nanotubes.