The Gaussian Approximation, proposed originally by Ottinger [J. Chem. Phys., 90, 463-473 (1989)] to account for the influence of fluctuations in hydrodynamic interactions in Rouse chains, is adapted here to derive a new mean-field approximation for the FENE spring force. This FENE-PG force law approximately accounts for spring-force fluctuations, which are neglected in the widely used FENE-P approximation. The Gaussian Approximation for hydrodynamic interactions is combined with the FENE-P and FENE-PG spring force approximations to obtain approximate models for finitely extensible bead-spring chains with hydrodynamic interactions. The closed set of ordinary differential equations governing the evolution of the second moments of the configurational probability distribution in the approximate models is used to generate predictions of rheological properties in steady and unsteady shear and uniaxial extensional flows, which are found to be in good agreement with the exact results obtained with Brownian dynamics simulations. In particular, predictions of coil-stretch hysteresis are in quantitative agreement with simulations' results. (c) 2006 The Society of Rheology.