This study presents a topology optimization method for thermal conductivity problems that incorporates thermal radiation boundary conditions depending on design variables. In the optimization process, partial differential equations (PDEs) are introduced to represent geometric features with high thermal radiation, and the solution is employed to numerically extract the thermal radiation boundary. The boundary conditions of the temperature governing equations are determined using the proposed mathematical model and used to solve the temperature field. An optimal shape is created by iterative computation based on the level set method. In this study, the finite element method (FEM) is used to solve the PDEs of the heat transfer problems, and the level set functions are updated. Numerical examples are presented to verify the effectiveness and practicality of the proposed method.