  The asymptotic structure of the QCD perturbative relation between the
on-shell and $\overline{\rm{MS}}$ heavy quark masses is studied. We estimate
the five and six-loop contributions to this relation by three different
techniques. First, the effective charges motivated approach in two variants is
used. Second, the results following from the large-$\beta_0$ approximation are
analyzed. Finally, the consequences of applying the asymptotic renormalon-based
formula are investigated. We show that all approaches lead to corrections which
are qualitatively consistent in order of magnitude. Their sign-alternating
character in powers of the number of massless quarks is demonstrated. We
emphasize that there is no contradiction in the behavior of the fine structure
of the renormalon-based estimates with other approaches if one use the detailed
information about the normalization factor included in the renormalon
asymptotic formula. The obtained five- and six-loop estimates indicate that in
the case of the $b$-quark the asymptotic character of the studied relation
manifests itself above the fourth order of PT, whereas for the $t$-quark it
starts to reveal itself after the seventh order. This allows to conclude that
like the running masses, the pole masses of the $b$ and especially $t$-quark in
principle may be used in the phenomenologically-oriented studies.
