A new numerical algorithm is presented to solve steady-state nonlinear heat conduction problems by combining the scaled boundary finite element method and the Kirchhoff transformation. In order to eliminate the nonlinearity related to the temperature dependence of the thermal conductivity, the Kirchhoff transformation is utilized to convert the nonlinear partial differential governing equation into the Laplace equation, and therefore iterative computation can be avoided. As a semi-analytical numerical method that combines the advantages of finite element method and boundary element method, the scaled boundary finite element method only requires the boundary without fundamental solution to be discretized. After the numerical solutions with high precision in the transformation space are calculated by the scaled boundary finite element method, the inverse transformation is required to derive the temperature field. Several numerical examples are presented to verify that the proposed method is effective for steady-state nonlinear heat conduction problem.