The flow of a viscoelastic fluid in a cross-slot flow is studied numerically for two different versions of the elastic-dumbbell model, namely, the Chilcott-Rallison model (FENE-CR) and the same model but with conformation-dependent friction modeled via the Hinch-de Gennes scaling (FENE-CD). It is shown that the FENE-CR and FENE-CD models both predict a local reduction in the streamwise velocity downstream of the stagnation point. The reduction in the velocity is shown to result from the presence of highly stretched dumbbells near the streamlines emanating from the stagnation point, and is in reasonable agreement with existing experimental observations on high molecular weight polymer solutions. However, the FENE-CR model is shown to produce a birefringent strand whose length in the streamwise direction is significantly shorter than observed experimentally. Better agreement with experimental observations is found with the FENE-CD model, which exhibits a much slower relaxation from the extended state, thus producing a birefringent strand which extends substantially farther downstream than for the FENE-CR model. The effect of a conformation-dependent friction coefficient similar to the one proposed by Larson [R.G. Larson, The unravelling of a polymer chain in a strong extensional flow Rheol. Acta 29 (1990) 371-384] is also considered.