We present an extension of the Lubachevsky and Stillinger [J. Stat. Phys. 60, 561 (1990)] packing algorithm to generate packings of polydisperse spheres. The original Lubachevsky-Stillinger algorithm is a nonequilibrium protocol that allows a set of monodisperse spheres to grow slowly over time eventually reaching an asymptotic maximum packing fraction. We use this protocol to pack polydisperse spheres in three dimensions by making the growth rate of a sphere proportional to its initial diameter. This allows us to specify a size distribution of spheres, which is then preserved throughout the growth process (except the mean diameter increases). We use this method to study the packing of bidisperse sphere systems in detail. The packing fractions of the configurations generated with our method are consistent with both previously generated experimental and simulated packings over a large range of volume ratios. Our modified Lubachevsky-Stillinger protocol, however, extends the range of sphere volume ratios well beyond that which has been previously considered using simulation. In doing so, it allows both small volume ratios and large volume ratios to be studied within a single framework. We also show that the modified Lubachevsky-Stillinger algorithm is appreciably more efficient than a recursive packing method.