This numerical study aims to investigate the capacity of the circle of Wills (CoW) to provide collateral blood supply for patients with unilateral carotid arterial stenosis. The basic 3D geometry of the CoW was reconstructed based on a magnetic resonance angiogram of a normal human subject. A total of 52 computational fluid dynamics simulations were performed for four geometry configurations of the CoW with an artificially inserted axisymmetric stenosis of different luminal area reductions in an internal carotid artery (ICA) under a variety of boundary conditions. The CoW geometric configurations included (a) a normal CoW with all communicating arteries; (b) as model (a) but with enlarged communicating arterial diameters; (c) as (a) but with the ipsilateral posterior communicating artery missing, and (d) as (c) but with enlarged communicating arteries. It is found that the blood perfusion pressure drop between the ipsilateral ICA and the middle cerebral artery (MCA) only becomes significant when the degree of stenosis is greater than 86%. The cerebral autoregulation range varied significantly between the different CoW configurations for the severe stenosis cases. Without causing the flow rates to decrease at the efferent arterial ends, the mean perfusion pressure in the ipsilateral ICA can drop from 100 to 73, 67, 92 and 84 mmHg for the CoW models (a)-(d) with 96% luminal area reduction stenosis, respectively. The additional pathways are able to raise the ipsilateral MCA pressure significantly without reducing the total flow perfusion. Cerebral autoregulation effects were not directly included in the study. Therefore, the findings in the study should be interpreted with cautions when comes to the biological and clinical significance.