In this paper we address the stationary Oldroyd model with diffusive stress in two ways: first we present the mathematical analysis of the model. Second, as the link between the diffusive model and the standard one is questionable from the mathematical point of view, we discuss, by means of numerical simulations, the behaviour of the model with respect to vanishing diffusion. In particular, numerical results in 2D or 3D suggest that the solution of the diffusive model converges to the solution of the non-diffusive model at order 1 in the $L^2-$norm and $H^1-$norm.