Eikonal equations are widely used in computer vision, image processing, geometric optics, etc. This paper extends the weighted compact nonlinear scheme (WCNS) and the weighted essentially non-oscillatory (WENO) scheme for hyperbolic conservation laws and designs high-order fast sweeping WCNS and WENO schemes to solve the pseudo-time dependent Eikonal equations. Fifth-order WCNS and WENO schemes are applied to compute the left and right limit values of spatial derivatives of the unknown variable coupled with the monotone Lax-Friedrichs numerical Hamiltonians. In order to speed up the convergence of the designed algorithm and to avoid solving a nonlinear discrete system, an explicit time-marching scheme combined with a fast sweeping strategy is used for time discretization. Numerical results show that both the fast sweeping WCNS method and the fast sweeping WENO method can achieve fifth-order accuracy in smooth regions and the numerical solutions obtained with the two methods are in good agreement with the exact solutions of Eikonal equations. Compared with the classical WENO method of the same order, the fast sweeping WCNS and WENO schemes are more efficient when they obtain the same numerical errors.