In order to describe the translational diffusion of polymers in dilute solution, an elastic dumbbell model is used for which the effects of both hydrodynamic interaction and a nonlinear connector force are treated in a self-consistently averaged form. For this model, the diffusion tensor in the presence of a homogeneous flow field is obtained from (i) the mass flux caused by concentration gradients, (ii) the average polymer velocity caused by external forces, and (iii) the mean-square displacement of a polymer caused by the Brownian forces. From the second and third approaches the same expression for the diffusion tensor is found, whereas a different expression is obtained from the first approach. This means that the Nernst-Einstein equation cannot be generalized to the case of flowing solutions. The model predictions for the diffusion tensors in steady shear flow are discussed in detail.