Steady state problems for nonlinear hyperbolic conservation laws widely exist in many fields, such as fluid mechanics, optimal control, picture processing, and so on. A fast sweeping weighted compact nonlinear scheme (WCNS) is designed for these problems. The model equations are first transformed into the time-dependent problems by adding time derivative terms. Then, the flux derivatives are computed by a third-order WCNS and a third-order TVD Runge-Kutta method is used for the time discretization. Moreover, the fast sweeping strategy that combines Gauss-Seidel-type iteration method with a finite number of alternating sweeping orderings is adopted to accelerate the convergence speed of the designed algorithm. Numerical results show that the fast sweeping WCNS has third-order accuracy and good shock-capturing ability. Compared with the TVD Runge-Kutta WCNS without sweeping, the fast sweeping WCNS has a faster convergence speed and higher computational efficiency.