Weyl semimetals (WSMs) have attracted the attention of the researchers due to their fascinating properties which are analogous to that of three-dimensional graphene. The density functional theory (under generalised gradient approximation (GGA) without spin-orbit coupling, GGA with spin-orbit coupling(GGA+SOC) and GGA with Hubbard correction (GGA+U)) in combination with the stress–strain approach have been utilised to investigate the elastic and mechanical properties of ZrS and ZrTe. The thermodynamic propertieshave been evaluated using the quasi-harmonic approximations by incorporating GGA, GGA+SOC and GGA+U approaches. The polycrystalline elastic moduli have been calculated using the single-crystal elastic constantsand the mechanical stabilities have also been established. Physical parameters, such as Young’s modulus, shear modulus, Poisson’s ratio, Debye temperature and sound velocities, are also calculated. In addition, the anisotropic elastic properties such as Young’s modulus, linear compressibility, shear modulus and Poisson’s ratio as well as the anisotropic factors have been visualised in three dimensions (3D) using GGA, GGA+SOC and GGA+U approaches. The theoretical computation of thermodynamic properties such as specific heat, entropy, vibration energy and internal energy as a function of temperature for both Weyl semimetals are investigated and discussed for the first time. The calculated values of Debye temperature for ZrS (ZrTe) are 470.103 K (287.744 K), 486.572K (298.295 K) and 442.0 K (234.346 K) using GGA approximations without SOC, with SOC and by implementing GGA+U calculations, respectively. Further, it is shown that the value of Debye temperature for ZrS ismore than thatof ZrTe. Hence, the present study of thermodynamic properties suggests their potential thermoelectric applications at high temperatures.