In Hu and Yi (2016) [20], a non-oscillatory k-exact reconstruction method was proposed towards the high-order finite volume methods for steady Euler equations, which successfully demonstrated the high-order behavior in the simulations. However, the degeneracy of the numerical accuracy of the approximate solutions to problems with curved boundary can be observed obviously. In this paper, the issue is resolved by introducing the Non-Uniform Rational B-splines (NURBS) method, i.e., with given discrete description of the computational domain, an approximate NURBS curve is reconstructed to provide quality quadrature information along the curved boundary. The advantages of using NURBS include i). both the numerical accuracy of the approximate solutions and convergence rate of the numerical methods are improved simultaneously, and ii). the NURBS curve generation is independent of other modules of the numerical framework, which makes its application very flexible. It is also shown in the paper that by introducing more elements along the normal direction for the reconstruction patch of the boundary element, significant improvement in the convergence to steady state can be achieved. The numerical examples confirm the above features very well.