Last decade has seen a surge in the development and deployment of low Earth orbit (LEO) constellations primarily serving broadband communications applications. These developments have influenced the interest in providing positioning, navigation, and timing (PNT) services from LEO, thus aiming to augment the well-established global navigation satellite systems (GNSSs) in medium Earth orbit (MEO) by new signals and services. Yet another proposal is to utilize LEO constellations to monitor and detect faults in GNSSs. This approach promises an advantage over existing ground-based monitoring, primarily due to minimized atmospheric impact in the observations. In this paper we investigate the influence of LEO constellation design on the line-of-sight (LOS) visibility conditions for GNSS monitoring. We simulate a series of Walker constellations in LEO with varying number of total satellites, orbital planes, and orbital heights. From the simulated data we gather statistics on number of visible satellites, durations of visibilities, and the quality of this visibility quantified by the dilution of precision (DOP) metric. Our findings indicate a relatively small number of satellites sufficing in providing an adequate monitoring capability. We also identify orbital geometric constraints resulting in suboptimal performance and provide optimization strategies.