Thermal interface material (TIM) that can exist as liquid at the service temperature enables efficient heat transfer across two adjacent surfaces in electronic applications. In this work, the thermal conductivities of different phase regions in the Ga-In system at various compositions and temperatures are measured for the first time. A modified comparative cut bar technique is used for the measurement of the thermal conductivities of In_xGa_1-x (x=0, 0.1, 0.214, 0.3, and 0.9) alloys at 40, 60, 80, and 100^oC that are the temperatures commonly encountered in consumer electronics. The thermal conductivity values for the liquid and semi-liquid (liquid+β) Ga-In alloys are higher than the TIM currently used in consumer electronics. These measured quantities, along with the available experimental data from the literature, served as input for the thermal conductivity parameter optimization using the CALPHAD (CALculation of PHase Diagram) method for the pure elements, solution phase, and two-phase region. A set of self-consistent parameters for the description of the thermal conductivity of the Ga-In system is obtained. There is good agreement between the measured and calculated thermal conductivity values for all the phases. Hence, it can be envisaged that liquid/semi-liquid Ga-In alloys can be considered as a potential TIM in consumer electronics due to its high thermal conductivity.