In Li and Ren (Int. J. Numer. Methods Fluids 70: 742–763, 2012), a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain, in which the high-order numerical accuracy and the oscillations-free property can be achieved. In this paper, the method is extended to solve steady state problems imposed in a curved physical domain. The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations, and a geometrical multigrid method to solve the derived linear system. To achieve high-order non-oscillatory numerical solutions, the classical k-exact reconstruction with 𝑘=3k=3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables. The non-uniform rational B-splines (NURBS) curve is used to provide an exact or a high-order representation of the curved wall boundary. Furthermore, an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state. A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.