The Solar Probe Plus (SPP) mission will explore the Sun's corona and innermost solar wind starting in 2018. The spacecraft will also come close to a number of Mercurycrossing asteroids with perihelia less than 0.3 AU. At small heliocentric distances, these objects may begin to lose mass, thus becoming "active asteroids" with comet-like comae or tails. This paper assembles a database of 97 known Mercury-crossing asteroids that may be encountered by SPP, and it presents estimates of their time-dependent visible-light fluxes and mass loss rates. Assuming a similar efficiency of sky background subtraction as was achieved by STEREO, we find that approximately 80% of these asteroids are bright enough to be observed by the Wide-field Imager for SPP (WISPR). A model of gas/dust mass loss from these asteroids is developed and calibrated against existing observations. This model is used to estimate the visible-light fluxes and spatial extents of spherical comae. Observable dust clouds occur only when the asteroids approach the Sun closer than 0.2 AU. The model predicts that during the primary SPP mission between 2018 and 2025, there should be 113 discrete events (for 24 unique asteroids) during which the modeled comae have angular sizes resolvable by WISPR. The largest of these correspond to asteroids 3200 Phaethon, 137924, 155140, and 289227, all with angular sizes of roughly 15 to 30 arcminutes. We note that the SPP trajectory may still change, but no matter the details there should still be multiple opportunities for fruitful asteroid observations.