Abstract Numerical simulations have been undertaken for the creeping flow of two well-characterized polymer solutions (fluids W1 and S1) past a sphere in a tightly-fitting cylindrical tube with a 1.14:1 diameter ratio. The fluids have been modelled using an integral constitutive equation of the K-BKZ type with a spectrum of relaxation times. Numerical values for the constants appearing in the equation have been obtained from fitting shear viscosity and normal stress data as measured in shear. Convergence of the numerical scheme was obtained for the whole range of experiments by using the adaptive viscoelastic stress splitting (AVSS) technique. The numerical solutions show that the drag on the sphere decreases dramatically for the shear-thinning S1 solution, while it decreases by about 20% for the constant-viscosity W1 solution. The results are in very good quantitative agreement with previous experimental findings for low-to-moderate Weissenberg numbers, but provide underestimates reaching about 10% in the high elasticity range. The latter is seen to be caused by sensitivity of the results to the rheological properties and their fitting at high shear rates.