This work examines the heat transfer between two fixed contacting specimens in order to estimate the time-varying thermal contact conductance. An inverse heat conduction method is presented. A well-known technique called extrapolation method is also utilized to obtain the results by using data taken from an experiment. The results achieved by the inverse and linear extrapolation methods are in good agreement. In addition, a steady-state analysis has been performed by using the commercial software based on the finite element method and an error analysis shows that the extrapolation method is superior in this analysis.