Casting is considered as a major manufacturing process. Thermal analysis of a solidifying medium is of great importance for appropriate design of casting processes. The conventional governing equation of a solidifying medium is based on the Fourier heat conduction law, which does not account for the phase-lag between the heat flux and the temperature gradient. In this paper, the concept of phase-lag during the phenomenon of solidification is investigated. This concept is considered by utilization of the hyperbolic heat conduction equation, known generally as the Maxwell–Cattaneo relation. In this way, the effect of finite heat wave speed on the thermal behavior of a solidifying medium is studied. In this context, some numerical example problems are analyzed with the meshless radial point interpolation method. The effect of the relaxation time on the thermal behavior of the solidifying medium is investigated. Moreover, the results of Fourier and non-Fourier heat conduction equations are compared. It is observed that based on the specific solidification process and the amount of relaxation time, the results of the Fourier and non-Fourier conduction laws can be quite different. The most prominent effect of the relaxation time is to alter the initiation of the solidification at each point.