Long-time tails of the molecular velocity autocorrelation function (VACF) in liquid argon at temperatures higher and lower than the spinodal temperature have been analyzed. By considering the time dependence of the VACF, the self-diffusion and shear viscosity coefficients, and the Maxwell relaxation time are determined, as well as their changes when crossing the spinodal. It is shown that the characteristic changes in the temperature dependences of the indicated kinetic coefficients allow the spinodal position to be determined with a high accuracy. A possibility to apply the proposed method to other low-molecular liquids is considered. As an example, nitrogen and oxygen are used, for which the averaged potential of intermolecular interaction has the Lennard-Jones form.K e y w o r d s: self-diffusion coefficient, kinematic shear viscosity coefficient, Maxwell relaxation time, argon spinodal, averaged potential of molecular interaction.