The under-abundance of asteroids on orbits with small perihelion distances suggests that thermally driven disruption may be an important process in the removal of rocky bodies in the solar system. Here we report our study of how the debris streams arise from possible thermally driven disruptions in the near-Sun region. We calculate that a small body with a diameter 0.5 km can produce a sufficient amount of material to allow the detection of the debris at the Earth as meteor showers, and that bodies at such sizes thermally disrupt every ~2 kyr. We also find that objects from the inner parts of the asteroid belt are more likely to become Sun-approachers than those from the outer parts. We simulate the formation and evolution of the debris streams produced from a set of synthetic disrupting asteroids drawn from Granvik et al.'s near-Earth object population model, and find that they evolve 10–70 times faster than streams produced at ordinary solar distances. We compare the simulation results to a catalog of known meteor showers on Sun-approaching orbits. We show that there is a clear overabundance of Sun-approaching meteor showers, which is best explained by a combining effect of comet contamination and an extended disintegration phase that lasts up to a few thousand years. We suggest that a few asteroid-like Sun-approaching objects that brighten significantly at their perihelion passages could, in fact, be disrupting asteroids. An extended period of thermal disruption may also explain the widespread detection of transiting debris in exoplanetary systems.