ABSTRACT The observed near-Earth asteroid (NEA) population contains very few objects with small perihelion distances, say, $q \lesssim 0.2\, \mathrm{au}$. NEAs that currently have orbits with larger q might be hiding a past evolution during which they have approached closer to the Sun. We present a probabilistic assessment of the minimum q that an asteroid has reached during its orbital history. At the same time, we offer an estimate of the dwell time, that is, the time q has been in a specific range. We have re-analysed orbital integrations of test asteroids from the moment they enter the near-Earth region until they either collide with a major body or are thrown out from the inner Solar system. We considered a total disruption of asteroids at certain q as a function of absolute magnitude (H). We calculated the probability that an asteroid with given orbital elements and H has reached a q smaller than a given threshold value and its respective dwell time in that range. We have constructed a look-up table that can be used to study the past orbital and thermal evolution of asteroids as well as meteorite falls and their possible parent bodies. An application to 25 meteorite falls shows that carbonaceous chondrites typically have short dwell times at small q, whereas for ordinary chondrites it ranges from $10\, 000$ to $500\, 000$ yr. A dearth of meteorite falls with long dwell times and small minimum q supports a supercatastrophic disruption of asteroids at small q.