Recent simulations have indicated that the dark matter halos of galaxy clusters should feature steep density jumps near the virial radius. Since the member galaxies are expected to follow similar collisionless dynamics as the dark matter, the galaxy density profile should show such a feature as well. We examine the potential of current data sets to test this prediction by selecting cluster members for a sample of 56 low-redshift (0.1\lt z\lt 0.3) galaxy clusters, constructing their projected number density profiles, and fitting them with two profiles, one with a steep density jump and one without. Additionally, we investigate the presence of a jump using a non-parametric spline approach. We find that some of these clusters show strong evidence for a model with a density jump, with the strength of the signal increasing with the inclusion of spectroscopic cluster member identification. We discuss avenues for further analysis of the density jump with future data sets, particularly with the inclusion of additional spectroscopy of cluster outskirts.