In this paper we consider a Proper Generalized Decomposition method to solve the steady incompressible Navier–Stokes equations with random Reynolds number and forcing term. The aim of such technique is to compute a low-cost reduced basis approximation of the full Stochastic Galerkin solution of the problem at hand. A particular algorithm, inspired by the Arnoldi method for solving eigenproblems, is proposed for an efficient greedy construction of a deterministic reduced basis approximation. This algorithm decouples the computation of the deterministic and stochastic components of the solution, thus allowing reuse of pre-existing deterministic Navier–Stokes solvers. It has the remarkable property of only requiring the solution of m uncoupled deterministic problems for the construction of a m-dimensional reduced basis rather than M coupled problems of the full Stochastic Galerkin approximation space, with m << M (up to one order of magnitude for the problem at hand in this work).