We compute numerically the steady flow of a viscoelastic FENE-P fluid in a two-dimensional collapsible channel, in which a zero-thickness membrane under constant tension has replaced part of the top wall. This geometry has been studied extensively as a benchmark to examine fluid-structure interactions between a flowing Newtonian fluid and a deformable wall. The theological behavior of the viscoelastic FENE-P fluid is described in terms of a conformation tensor model. The mesh equation and transport equations are discretized using the discrete elastic viscous stress split-traceless gradient (DEVSS-TG/SUPG) mixed finite-element method. The influence of viscoelasticity and shear thinning on flow patterns and stress profiles is examined, and comparisons with Newtonian predictions are reported. The existence of a limiting Weissenberg number beyond which computations fail is demonstrated. The extent of shear thinning is shown to be a key factor in determining the nature of the fluid-structure interaction.