Summary: The linear space of all proper rational functions with prescribed poles is considered. Given a set of points z\(_{i}\) in the complex plane and the weights \(w_i\) we define the discrete inner product \[ \langle \phi,\psi \rangle := \sum_{i=0}^n |w_i|^2 \overline{\phi(z_i)} \psi(z_i). \] We derive a method to compute the coefficients of a recurrence relation generating a set of orthonormal rational basis functions with respect to the discrete inner product. We show that these coefficients can be computed by solving an inverse eigenvalue problem for a matrix having a specific structure. In the case where all the points \(z_i\) lie on the real line or on the unit circle, the computational complexity is reduced by an order of magnitude.