In this article we study the numerical approximation of an Oseen type model for viscoelastic fluid flow. Existence and uniqueness of the continuous and approximate solutions, under a small data assumption, are proved. Error estimates for the numerical approximations are also derived. Numerical experiments are presented which support the error estimates, and which demonstrate the relevance of the small data assumption for the solvability of the continuous and discrete systems. (c) 2005 Elsevier B.V. All rights reserved.