We address the closure problem for the most elementary non-linear kinetic model of a dilute polymeric solution, known as the Warner finitely extensible non-linear elastic (FENE) dumbbell model. In view of the closure problem, the FENE theory cannot be translated into an equivalent macroscopic constitutive equation for the polymer contribution to the stress tensor. We present a general framework for developing closure approximations, based on the concept of canonical distribution subspace first introduced by Verleye and Dupret (in: Developments in Non-Newtonian Flows, AMD-Vol. 175, ASME, New York, 1993, 139-163) in the context of fiber suspension modeling. The classical consistent pre-averaging approximation due to Peterlin (that yields the FENE-P constitutive equation) is obtained from the canonical approach as the simplest first-order closure model involving only the second moment of the configuration distribution function. A second-order closure model (referred to as FENE-P-2) is derived, which involves the second and fourth moments of the distribution function. We show that the FENE-P-2 model behaves like the FENE-P equation with a reduced extensibility parameter. In this respect, it is a close relative of the FENE-P* equation proposed by van Heel et al. (J. Non-Newton. Fluid Mech., 1998, in press). Inspired by stochastic simulation results for the FENE theory, we propose a more sophisticated second-order closure model (referred to as FENE-L). The rheological response of the FENE-P, FENE-P-2 and FENE-L closure models are compared to that of the FENE theory in various time-dependent, one-dimensional elongational flows. Overall, the FENE-L model is found to provide the best agreement with the FENE results. In particular, it is capable of reproducing the hysteretic behaviour of the FENE model, also observed in recent experiments involving polystyrene-based Boger fluids (Doyle et al., J. Non-Newton. Fluid Mech., submitted), in stress versus birefringence curves during startup of flow and subsequent relaxation.