A question whether thermal convection occurs in the solid inner core or not is not yet completely solved. In this paper the temperature profile in the core is examined when the solid core grows, being cooled from the surface. In order for convection to occur, the temperature gradient must exceed the adiabatic gradient. In the solid inner core, temperature due to thermal conduction is calculated and compared with the adiabatic temperature. In the outer core, temperature is assumed to be adiabatic due to vigorous fluid convection. The temperature gradient in the solid inner core has been estimated under different conditions, i.e., for different growth rates and different concentration of radioactive material, but it has been found that it can never exceed the adiabatic gradient. If we assume that the inner core has been solidified to the present size in 4.5 billion years, the temperature decreases, maintained almost isothermal inside the inner core. Even if it has been formed in such a short period as 1.5 billion years, the temperature gradient is still much smaller than the adiabat. This indicates that no thermal convection can occur. Presence of radiogenic heat sources raises the temperature, but even with such a high concentration as 100 ppm of K no convection is likely to occur.