For numerically simulating the melt behavior in the core catcher of a sodium-cooled reactor, the HEFEST-FR module—a software tool based on the SAFR computer code elaborated at the Nuclear Safety Institute, Russian Academy of Sciences, for simulating the meltdown and destruction of liquid metal-cooled fast reactor core components—has been developed and incorporated into the EUCLID/V2 integrated computer code. This module is intended for numerically simulating the melt retention and cooling-down processes in the reactor vessel with taking into account the heat transfer from the vessel’s internal structures to the coolant. For this purpose, a 2D-problem (implying that there is no dependence of temperature on the azimuthal angle) of unsteady heat conduction for materials located in the reactor’s core catcher tray is solved in a cylindrical coordinate system. The heat-conduction equation coefficients depend on time, coordinates, and temperature, the latter being the solution of equation. Boundary conditions of the first, second, and third kind are used, and heat loss by radiation at the boundary is specified. The total or volumetric decay heat power is given inside of the melt. For numerically solving the 2D heat-conductivity problem, an enthalpy approach based method is applied. The formulations used in the method make it possible to overcome, in a natural manner, the problem relating to discontinuity of material specific melting point enthalpy in solving the heat-conduction problem with taking phase transitions into account. The solution yields the temperature field in melting/freezing of materials (steels of various grades and fuel) in the fast reactor’s core catcher tray. The results of verifying the HEFEST-FR module against the solution of an analytical problem have been demonstrated. Using the module, methodical computations of the fuel and fuel pin cladding melt behavior in the sodium-cooled reactor’s core catcher have been carried out.