Large regions of suppressed recombination having extended over time occur in many organisms around genes involved in mating compatibility (sex-determining or mating-type genes). The sheltering of deleterious alleles has been proposed to be involved in such expansions. However, the dynamics of deleterious mutations partially linked to genes involved in mating compatibility are not well understood, especially in finite populations. In particular, under what conditions deleterious mutations are likely to be maintained for long enough near mating-compatibility genes remains to be evaluated, especially under selfing, which generally increases the purging rate of deleterious mutations. Using a branching process approximation, we studied the fate of a new deleterious or overdominant mutation in a diploid population, considering a locus carrying two permanently heterozygous mating-type alleles, and a partially linked locus at which the mutation appears. We obtained analytical and numerical results on the probability and purging time of the new mutation. We investigated the impact of recombination between the two loci and of the mating system (outcrossing, intra and inter-tetrad selfing) on the maintenance of the mutation. We found that the presence of a fungal-like mating-type locus (i.e. not preventing diploid selfing) always sheltered the mutation under selfing, i.e. it decreased the purging probability and increased the purging time of the mutations. The sheltering effect was higher in case of automixis (intra-tetrad selfing). This may contribute to explain why evolutionary strata of recombination suppression near the mating-type locus are found mostly in automictic (pseudo-homothallic) fungi. We also showed that rare events of deleterious mutation maintenance during strikingly long evolutionary times could occur, suggesting that deleterious mutations can indeed accumulate near the mating-type locus over evolutionary time scales. In conclusion, our results show that, although selfing purges deleterious mutations, these mutations can be maintained for very long times near a mating-type locus, which may contribute to promote the evolution of recombination suppression in sex-related chromosomes.