The propagation of monochromatic electromagnetic TM waves in the Goubau line (conducting cylinder covered by a concentric dielectric layer) filled with nonlinear inhomogeneous medium is considered. Two types of nonlinearity are studied. A physical problem is reduced to solving a nonlinear transmission eigenvalue problem for a system of ordinary differential equations. Eigenvalues of the problem correspond to propagation constants of the waveguide. A method is proposed for finding approximate eigenvalues of the nonlinear problem based on solving an auxiliary Cauchy problem (by the shooting method). The existence of eigenvalues that correspond to a new propagation regime is predicted. A comparison with the linear case is given.