Nonlinear coupled electromagnetic TE-TM wave propagation in the Goubau line (a conducting cylinder covered by a concentric dielectric layer) filled with nonlinear inhomogeneous medium is considered. Nonlinearity inside the GL is described by the Kerr law. The physical problem is reduced to a nonlinear two-parameter eigenvalue problem for a system of (nonlinear) ordinary differential equations. For the numerical solution, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed. The coupled TE-TM waves propagating in GL are determined numerically. Whether these mathematically predicted propagation regime really exist is a hypothesis that can be proved or disproved in an experiment.